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<!DOCTYPE html>
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<title>Chapter 22 The t-test | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
<meta name="description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
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<meta property="og:title" content="Chapter 22 The t-test | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
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<meta name="twitter:title" content="Chapter 22 The t-test | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
<meta name="twitter:description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
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<meta name="author" content="A. Bradley Duthie" />
<meta name="date" content="2024-08-06" />
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<ul class="summary">
<li><a href="./">Statistics with jamovi</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#structure"><i class="fa fa-check"></i>How this book is structured</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#datasets"><i class="fa fa-check"></i>Datasets used in this book</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#acknowledgements"><i class="fa fa-check"></i>Acknowledgements</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#author"><i class="fa fa-check"></i>About the author</a></li>
</ul></li>
<li class="chapter" data-level="1" data-path="Chapter_1.html"><a href="Chapter_1.html"><i class="fa fa-check"></i><b>1</b> Background mathematics</a>
<ul>
<li class="chapter" data-level="1.1" data-path="Chapter_1.html"><a href="Chapter_1.html#numbers-and-operations"><i class="fa fa-check"></i><b>1.1</b> Numbers and operations</a></li>
<li class="chapter" data-level="1.2" data-path="Chapter_1.html"><a href="Chapter_1.html#logarithms"><i class="fa fa-check"></i><b>1.2</b> Logarithms</a></li>
<li class="chapter" data-level="1.3" data-path="Chapter_1.html"><a href="Chapter_1.html#order-of-operations"><i class="fa fa-check"></i><b>1.3</b> Order of operations</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="Chapter_2.html"><a href="Chapter_2.html"><i class="fa fa-check"></i><b>2</b> Data organisation</a>
<ul>
<li class="chapter" data-level="2.1" data-path="Chapter_2.html"><a href="Chapter_2.html#tidy-data"><i class="fa fa-check"></i><b>2.1</b> Tidy data</a></li>
<li class="chapter" data-level="2.2" data-path="Chapter_2.html"><a href="Chapter_2.html#data-files"><i class="fa fa-check"></i><b>2.2</b> Data files</a></li>
<li class="chapter" data-level="2.3" data-path="Chapter_2.html"><a href="Chapter_2.html#managing-data-files"><i class="fa fa-check"></i><b>2.3</b> Managing data files</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="Chapter_3.html"><a href="Chapter_3.html"><i class="fa fa-check"></i><b>3</b> <em>Practical</em>. Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Tidy data and spreadsheet calculations</a></li>
<li class="chapter" data-level="3.5" data-path="Chapter_3.html"><a href="Chapter_3.html#summary"><i class="fa fa-check"></i><b>3.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="Chapter_4.html"><a href="Chapter_4.html"><i class="fa fa-check"></i><b>4</b> Populations and samples</a></li>
<li class="chapter" data-level="5" data-path="Chapter_5.html"><a href="Chapter_5.html"><i class="fa fa-check"></i><b>5</b> Types of variables</a></li>
<li class="chapter" data-level="6" data-path="Chapter_6.html"><a href="Chapter_6.html"><i class="fa fa-check"></i><b>6</b> Accuracy, precision, and units</a>
<ul>
<li class="chapter" data-level="6.1" data-path="Chapter_6.html"><a href="Chapter_6.html#accuracy"><i class="fa fa-check"></i><b>6.1</b> Accuracy</a></li>
<li class="chapter" data-level="6.2" data-path="Chapter_6.html"><a href="Chapter_6.html#precision"><i class="fa fa-check"></i><b>6.2</b> Precision</a></li>
<li class="chapter" data-level="6.3" data-path="Chapter_6.html"><a href="Chapter_6.html#systems-of-units"><i class="fa fa-check"></i><b>6.3</b> Systems of units</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="Chapter_7.html"><a href="Chapter_7.html"><i class="fa fa-check"></i><b>7</b> Uncertainty propagation</a>
<ul>
<li class="chapter" data-level="7.1" data-path="Chapter_7.html"><a href="Chapter_7.html#adding-or-subtracting-errors"><i class="fa fa-check"></i><b>7.1</b> Adding or subtracting errors</a></li>
<li class="chapter" data-level="7.2" data-path="Chapter_7.html"><a href="Chapter_7.html#multiplying-or-dividing-errors"><i class="fa fa-check"></i><b>7.2</b> Multiplying or dividing errors</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="Chapter_8.html"><a href="Chapter_8.html"><i class="fa fa-check"></i><b>8</b> <em>Practical</em>. Introduction to jamovi</a>
<ul>
<li class="chapter" data-level="8.1" data-path="Chapter_8.html"><a href="Chapter_8.html#summary_statistics_02"><i class="fa fa-check"></i><b>8.1</b> Summary statistics</a></li>
<li class="chapter" data-level="8.2" data-path="Chapter_8.html"><a href="Chapter_8.html#transforming_variables_02"><i class="fa fa-check"></i><b>8.2</b> Transforming variables</a></li>
<li class="chapter" data-level="8.3" data-path="Chapter_8.html"><a href="Chapter_8.html#computing_variables_02"><i class="fa fa-check"></i><b>8.3</b> Computing variables</a></li>
<li class="chapter" data-level="8.4" data-path="Chapter_8.html"><a href="Chapter_8.html#summary-1"><i class="fa fa-check"></i><b>8.4</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="Chapter_9.html"><a href="Chapter_9.html"><i class="fa fa-check"></i><b>9</b> Decimal places, significant figures, and rounding</a>
<ul>
<li class="chapter" data-level="9.1" data-path="Chapter_9.html"><a href="Chapter_9.html#decimal-places-and-significant-figures"><i class="fa fa-check"></i><b>9.1</b> Decimal places and significant figures</a></li>
<li class="chapter" data-level="9.2" data-path="Chapter_9.html"><a href="Chapter_9.html#rounding"><i class="fa fa-check"></i><b>9.2</b> Rounding</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="Chapter_10.html"><a href="Chapter_10.html"><i class="fa fa-check"></i><b>10</b> Graphs</a>
<ul>
<li class="chapter" data-level="10.1" data-path="Chapter_10.html"><a href="Chapter_10.html#histograms"><i class="fa fa-check"></i><b>10.1</b> Histograms</a></li>
<li class="chapter" data-level="10.2" data-path="Chapter_10.html"><a href="Chapter_10.html#barplots-and-pie-charts"><i class="fa fa-check"></i><b>10.2</b> Barplots and pie charts</a></li>
<li class="chapter" data-level="10.3" data-path="Chapter_10.html"><a href="Chapter_10.html#box-whisker-plots"><i class="fa fa-check"></i><b>10.3</b> Box-whisker plots</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="Chapter_11.html"><a href="Chapter_11.html"><i class="fa fa-check"></i><b>11</b> Measures of central tendency</a>
<ul>
<li class="chapter" data-level="11.1" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mean"><i class="fa fa-check"></i><b>11.1</b> The mean</a></li>
<li class="chapter" data-level="11.2" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mode"><i class="fa fa-check"></i><b>11.2</b> The mode</a></li>
<li class="chapter" data-level="11.3" data-path="Chapter_11.html"><a href="Chapter_11.html#the-median-and-quantiles"><i class="fa fa-check"></i><b>11.3</b> The median and quantiles</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="Chapter_12.html"><a href="Chapter_12.html"><i class="fa fa-check"></i><b>12</b> Measures of spread</a>
<ul>
<li class="chapter" data-level="12.1" data-path="Chapter_12.html"><a href="Chapter_12.html#the-range"><i class="fa fa-check"></i><b>12.1</b> The range</a></li>
<li class="chapter" data-level="12.2" data-path="Chapter_12.html"><a href="Chapter_12.html#the-inter-quartile-range"><i class="fa fa-check"></i><b>12.2</b> The inter-quartile range</a></li>
<li class="chapter" data-level="12.3" data-path="Chapter_12.html"><a href="Chapter_12.html#the-variance"><i class="fa fa-check"></i><b>12.3</b> The variance</a></li>
<li class="chapter" data-level="12.4" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-deviation"><i class="fa fa-check"></i><b>12.4</b> The standard deviation</a></li>
<li class="chapter" data-level="12.5" data-path="Chapter_12.html"><a href="Chapter_12.html#the-coefficient-of-variation"><i class="fa fa-check"></i><b>12.5</b> The coefficient of variation</a></li>
<li class="chapter" data-level="12.6" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-error"><i class="fa fa-check"></i><b>12.6</b> The standard error</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="Chapter_13.html"><a href="Chapter_13.html"><i class="fa fa-check"></i><b>13</b> Skew and kurtosis</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#skew"><i class="fa fa-check"></i><b>13.1</b> Skew</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#kurtosis"><i class="fa fa-check"></i><b>13.2</b> Kurtosis</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#moments"><i class="fa fa-check"></i><b>13.3</b> Moments</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="Chapter_14.html"><a href="Chapter_14.html"><i class="fa fa-check"></i><b>14</b> <em>Practical</em>. Plotting and statistical summaries in jamovi</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>14.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>14.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>14.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>14.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#comparing-across-sites"><i class="fa fa-check"></i><b>14.5</b> Comparing across sites</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="Chapter_15.html"><a href="Chapter_15.html"><i class="fa fa-check"></i><b>15</b> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#instructive-example"><i class="fa fa-check"></i><b>15.1</b> Instructive example</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#biological-applications"><i class="fa fa-check"></i><b>15.2</b> Biological applications</a></li>
<li class="chapter" data-level="15.3" data-path="Chapter_15.html"><a href="Chapter_15.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>15.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="15.4" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-distributions"><i class="fa fa-check"></i><b>15.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="Chapter_15.html"><a href="Chapter_15.html#binomial-distribution"><i class="fa fa-check"></i><b>15.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="15.4.2" data-path="Chapter_15.html"><a href="Chapter_15.html#poisson-distribution"><i class="fa fa-check"></i><b>15.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="15.4.3" data-path="Chapter_15.html"><a href="Chapter_15.html#uniform-distribution"><i class="fa fa-check"></i><b>15.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="15.4.4" data-path="Chapter_15.html"><a href="Chapter_15.html#normal-distribution"><i class="fa fa-check"></i><b>15.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="Chapter_15.html"><a href="Chapter_15.html#summary-2"><i class="fa fa-check"></i><b>15.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="Chapter_16.html"><a href="Chapter_16.html"><i class="fa fa-check"></i><b>16</b> Central Limit Theorem</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>16.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probability-and-z-scores"><i class="fa fa-check"></i><b>16.2</b> Probability and z-scores</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="Chapter_17.html"><a href="Chapter_17.html"><i class="fa fa-check"></i><b>17</b> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>17.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>17.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="17.3" data-path="Chapter_17.html"><a href="Chapter_17.html#central-limit-theorem"><i class="fa fa-check"></i><b>17.3</b> Central limit theorem</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="Chapter_18.html"><a href="Chapter_18.html"><i class="fa fa-check"></i><b>18</b> Confidence intervals</a>
<ul>
<li class="chapter" data-level="18.1" data-path="Chapter_18.html"><a href="Chapter_18.html#normal-distribution-cis"><i class="fa fa-check"></i><b>18.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="18.2" data-path="Chapter_18.html"><a href="Chapter_18.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>18.2</b> Binomial distribution CIs</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="Chapter_19.html"><a href="Chapter_19.html"><i class="fa fa-check"></i><b>19</b> The t-interval</a></li>
<li class="chapter" data-level="20" data-path="Chapter_20.html"><a href="Chapter_20.html"><i class="fa fa-check"></i><b>20</b> <em>Practical</em>. z- and t-intervals</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>20.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>20.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes"><i class="fa fa-check"></i><b>20.3</b> Confidence intervals for different sample sizes</a></li>
<li class="chapter" data-level="20.4" data-path="Chapter_20.html"><a href="Chapter_20.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>20.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="20.5" data-path="Chapter_20.html"><a href="Chapter_20.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>20.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="Chapter_21.html"><a href="Chapter_21.html"><i class="fa fa-check"></i><b>21</b> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>21.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>21.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>21.3</b> P-values, false positives, and power</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="Chapter_22.html"><a href="Chapter_22.html"><i class="fa fa-check"></i><b>22</b> The t-test</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> One sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test"><i class="fa fa-check"></i><b>22.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#paired-samples-t-test"><i class="fa fa-check"></i><b>22.3</b> Paired samples t-test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>22.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>22.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="22.5.1" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test"><i class="fa fa-check"></i><b>22.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.5.2" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>22.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="22.6" data-path="Chapter_22.html"><a href="Chapter_22.html#summary-3"><i class="fa fa-check"></i><b>22.6</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="Chapter_23.html"><a href="Chapter_23.html"><i class="fa fa-check"></i><b>23</b> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#one-sample-t-test-1"><i class="fa fa-check"></i><b>23.1</b> One sample t-test</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#paired-t-test"><i class="fa fa-check"></i><b>23.2</b> Paired t-test</a></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>23.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="23.4" data-path="Chapter_23.html"><a href="Chapter_23.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>23.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="23.5" data-path="Chapter_23.html"><a href="Chapter_23.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>23.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="Chapter_24.html"><a href="Chapter_24.html"><i class="fa fa-check"></i><b>24</b> Analysis of variance</a>
<ul>
<li class="chapter" data-level="24.1" data-path="Chapter_24.html"><a href="Chapter_24.html#f-distribution"><i class="fa fa-check"></i><b>24.1</b> F-distribution</a></li>
<li class="chapter" data-level="24.2" data-path="Chapter_24.html"><a href="Chapter_24.html#one-way-anova"><i class="fa fa-check"></i><b>24.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="24.2.1" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>24.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="24.2.2" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>24.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="24.2.3" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>24.2.3</b> ANOVA F-statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="24.3" data-path="Chapter_24.html"><a href="Chapter_24.html#assumptions-of-anova"><i class="fa fa-check"></i><b>24.3</b> Assumptions of ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="Chapter_25.html"><a href="Chapter_25.html"><i class="fa fa-check"></i><b>25</b> Multiple comparisons</a></li>
<li class="chapter" data-level="26" data-path="Chapter_26.html"><a href="Chapter_26.html"><i class="fa fa-check"></i><b>26</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="27" data-path="Chapter_27.html"><a href="Chapter_27.html"><i class="fa fa-check"></i><b>27</b> Two-way ANOVA</a></li>
<li class="chapter" data-level="28" data-path="Chapter_28.html"><a href="Chapter_28.html"><i class="fa fa-check"></i><b>28</b> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-site"><i class="fa fa-check"></i><b>28.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-profile"><i class="fa fa-check"></i><b>28.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#multiple-comparisons"><i class="fa fa-check"></i><b>28.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="28.4" data-path="Chapter_28.html"><a href="Chapter_28.html#kruskal-wallis-h-test"><i class="fa fa-check"></i><b>28.4</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="28.5" data-path="Chapter_28.html"><a href="Chapter_28.html#two-way-anova"><i class="fa fa-check"></i><b>28.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="Chapter_29.html"><a href="Chapter_29.html"><i class="fa fa-check"></i><b>29</b> Frequency and count data</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-distribution"><i class="fa fa-check"></i><b>29.1</b> Chi-square distribution</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-goodness-of-fit"><i class="fa fa-check"></i><b>29.2</b> Chi-square goodness of fit</a></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>29.3</b> Chi-square test of association</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="Chapter_30.html"><a href="Chapter_30.html"><i class="fa fa-check"></i><b>30</b> Correlation</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#scatterplots"><i class="fa fa-check"></i><b>30.1</b> Scatterplots</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-coefficient"><i class="fa fa-check"></i><b>30.2</b> Correlation coefficient</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="30.2.2" data-path="Chapter_30.html"><a href="Chapter_30.html#spearmans-rank-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.2</b> Spearman’s rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>30.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="31" data-path="Chapter_31.html"><a href="Chapter_31.html"><i class="fa fa-check"></i><b>31</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>31.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>31.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#chi-square-test-of-association-1"><i class="fa fa-check"></i><b>31.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>31.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#spearmans-rank-correlation-test"><i class="fa fa-check"></i><b>31.5</b> Spearman’s rank correlation test</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>31.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="chapter" data-level="32" data-path="Chapter_32.html"><a href="Chapter_32.html"><i class="fa fa-check"></i><b>32</b> Simple linear regression</a>
<ul>
<li class="chapter" data-level="32.1" data-path="Chapter_32.html"><a href="Chapter_32.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>32.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="32.2" data-path="Chapter_32.html"><a href="Chapter_32.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>32.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="32.3" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-coefficients"><i class="fa fa-check"></i><b>32.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="32.4" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-line-calculation"><i class="fa fa-check"></i><b>32.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="32.5" data-path="Chapter_32.html"><a href="Chapter_32.html#coefficient-of-determination"><i class="fa fa-check"></i><b>32.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="32.6" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-assumptions"><i class="fa fa-check"></i><b>32.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="32.7" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>32.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="32.7.1" data-path="Chapter_32.html"><a href="Chapter_32.html#overall-model-significance"><i class="fa fa-check"></i><b>32.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="32.7.2" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>32.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="32.7.3" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-slope"><i class="fa fa-check"></i><b>32.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="32.7.4" data-path="Chapter_32.html"><a href="Chapter_32.html#simple-regression-output"><i class="fa fa-check"></i><b>32.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="32.8" data-path="Chapter_32.html"><a href="Chapter_32.html#prediction-with-linear-models"><i class="fa fa-check"></i><b>32.8</b> Prediction with linear models</a></li>
<li class="chapter" data-level="32.9" data-path="Chapter_32.html"><a href="Chapter_32.html#conclusion"><i class="fa fa-check"></i><b>32.9</b> Conclusion</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="Chapter_33.html"><a href="Chapter_33.html"><i class="fa fa-check"></i><b>33</b> Multiple regression</a>
<ul>
<li class="chapter" data-level="33.1" data-path="Chapter_33.html"><a href="Chapter_33.html#adjusted-coefficient-of-determination"><i class="fa fa-check"></i><b>33.1</b> Adjusted coefficient of determination</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="Chapter_34.html"><a href="Chapter_34.html"><i class="fa fa-check"></i><b>34</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="34.1" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-soil-depth"><i class="fa fa-check"></i><b>34.1</b> Predicting pyrogenic carbon from soil depth</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-fire-frequency"><i class="fa fa-check"></i><b>34.2</b> Predicting pyrogenic carbon from fire frequency</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_34.html"><a href="Chapter_34.html#multiple-regression-depth-and-fire-frequency"><i class="fa fa-check"></i><b>34.3</b> Multiple regression depth and fire frequency</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_34.html"><a href="Chapter_34.html#large-multiple-regression"><i class="fa fa-check"></i><b>34.4</b> Large multiple regression</a></li>
<li class="chapter" data-level="34.5" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-temperature-from-fire-frequency"><i class="fa fa-check"></i><b>34.5</b> Predicting temperature from fire frequency</a></li>
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<li class="chapter" data-level="35" data-path="Chapter_35.html"><a href="Chapter_35.html"><i class="fa fa-check"></i><b>35</b> Randomisation</a>
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<li class="chapter" data-level="35.1" data-path="Chapter_35.html"><a href="Chapter_35.html#summary-of-parametric-hypothesis-testing"><i class="fa fa-check"></i><b>35.1</b> Summary of parametric hypothesis testing</a></li>
<li class="chapter" data-level="35.2" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-approach"><i class="fa fa-check"></i><b>35.2</b> Randomisation approach</a></li>
<li class="chapter" data-level="35.3" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-for-hypothesis-testing"><i class="fa fa-check"></i><b>35.3</b> Randomisation for hypothesis testing</a></li>
<li class="chapter" data-level="35.4" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-assumptions"><i class="fa fa-check"></i><b>35.4</b> Randomisation assumptions</a></li>
<li class="chapter" data-level="35.5" data-path="Chapter_35.html"><a href="Chapter_35.html#bootstrapping"><i class="fa fa-check"></i><b>35.5</b> Bootstrapping</a></li>
<li class="chapter" data-level="35.6" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-conclusions"><i class="fa fa-check"></i><b>35.6</b> Randomisation conclusions</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA.html"><a href="appendexA.html"><i class="fa fa-check"></i><b>A</b> Answers to chapter exercises</a>
<ul>
<li class="chapter" data-level="A.1" data-path="appendexA.html"><a href="appendexA.html#chapter-3"><i class="fa fa-check"></i><b>A.1</b> Chapter 3</a>
<ul>
<li class="chapter" data-level="A.1.1" data-path="appendexA.html"><a href="appendexA.html#exercise-3.1"><i class="fa fa-check"></i><b>A.1.1</b> Exercise 3.1:</a></li>
<li class="chapter" data-level="A.1.2" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2"><i class="fa fa-check"></i><b>A.1.2</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.3" data-path="appendexA.html"><a href="appendexA.html#exercise-3.3"><i class="fa fa-check"></i><b>A.1.3</b> Exercise 3.3</a></li>
<li class="chapter" data-level="A.1.4" data-path="appendexA.html"><a href="appendexA.html#exercise-3.4"><i class="fa fa-check"></i><b>A.1.4</b> Exercise 3.4</a></li>
</ul></li>
<li class="chapter" data-level="A.2" data-path="appendexA.html"><a href="appendexA.html#chapter-8"><i class="fa fa-check"></i><b>A.2</b> Chapter 8</a>
<ul>
<li class="chapter" data-level="A.2.1" data-path="appendexA.html"><a href="appendexA.html#exercise-8.1"><i class="fa fa-check"></i><b>A.2.1</b> Exercise 8.1</a></li>
<li class="chapter" data-level="A.2.2" data-path="appendexA.html"><a href="appendexA.html#exercise-8.2"><i class="fa fa-check"></i><b>A.2.2</b> Exercise 8.2</a></li>
<li class="chapter" data-level="A.2.3" data-path="appendexA.html"><a href="appendexA.html#exercise-8.3"><i class="fa fa-check"></i><b>A.2.3</b> Exercise 8.3</a></li>
</ul></li>
<li class="chapter" data-level="A.3" data-path="appendexA.html"><a href="appendexA.html#chapter-14"><i class="fa fa-check"></i><b>A.3</b> Chapter 14</a>
<ul>
<li class="chapter" data-level="A.3.1" data-path="appendexA.html"><a href="appendexA.html#exercise-14.1"><i class="fa fa-check"></i><b>A.3.1</b> Exercise 14.1</a></li>
<li class="chapter" data-level="A.3.2" data-path="appendexA.html"><a href="appendexA.html#exercise-14.2"><i class="fa fa-check"></i><b>A.3.2</b> Exercise 14.2</a></li>
<li class="chapter" data-level="A.3.3" data-path="appendexA.html"><a href="appendexA.html#exercise-14.3"><i class="fa fa-check"></i><b>A.3.3</b> Exercise 14.3</a></li>
<li class="chapter" data-level="A.3.4" data-path="appendexA.html"><a href="appendexA.html#exercise-14.4"><i class="fa fa-check"></i><b>A.3.4</b> Exercise 14.4</a></li>
<li class="chapter" data-level="A.3.5" data-path="appendexA.html"><a href="appendexA.html#exercise-14.5"><i class="fa fa-check"></i><b>A.3.5</b> Exercise 14.5</a></li>
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<li class="chapter" data-level="A.4" data-path="appendexA.html"><a href="appendexA.html#chapter-17"><i class="fa fa-check"></i><b>A.4</b> Chapter 17</a>
<ul>
<li class="chapter" data-level="A.4.1" data-path="appendexA.html"><a href="appendexA.html#exercise-17.1"><i class="fa fa-check"></i><b>A.4.1</b> Exercise 17.1</a></li>
<li class="chapter" data-level="A.4.2" data-path="appendexA.html"><a href="appendexA.html#exercise-17.2"><i class="fa fa-check"></i><b>A.4.2</b> Exercise 17.2</a></li>
<li class="chapter" data-level="A.4.3" data-path="appendexA.html"><a href="appendexA.html#exercise-17.3"><i class="fa fa-check"></i><b>A.4.3</b> Exercise 17.3</a></li>
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<li class="chapter" data-level="A.5" data-path="appendexA.html"><a href="appendexA.html#chapter-20"><i class="fa fa-check"></i><b>A.5</b> Chapter 20</a>
<ul>
<li class="chapter" data-level="A.5.1" data-path="appendexA.html"><a href="appendexA.html#exercise-20.1"><i class="fa fa-check"></i><b>A.5.1</b> Exercise 20.1</a></li>
<li class="chapter" data-level="A.5.2" data-path="appendexA.html"><a href="appendexA.html#exercise-20.2"><i class="fa fa-check"></i><b>A.5.2</b> Exercise 20.2</a></li>
<li class="chapter" data-level="A.5.3" data-path="appendexA.html"><a href="appendexA.html#exercise-20.3"><i class="fa fa-check"></i><b>A.5.3</b> Exercise 20.3</a></li>
<li class="chapter" data-level="A.5.4" data-path="appendexA.html"><a href="appendexA.html#exercise-20.4"><i class="fa fa-check"></i><b>A.5.4</b> Exercise 20.4</a></li>
<li class="chapter" data-level="A.5.5" data-path="appendexA.html"><a href="appendexA.html#exercise-20.5"><i class="fa fa-check"></i><b>A.5.5</b> Exercise 20.5</a></li>
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<li class="chapter" data-level="A.6" data-path="appendexA.html"><a href="appendexA.html#chapter-23"><i class="fa fa-check"></i><b>A.6</b> Chapter 23</a>
<ul>
<li class="chapter" data-level="A.6.1" data-path="appendexA.html"><a href="appendexA.html#exercise-23.1"><i class="fa fa-check"></i><b>A.6.1</b> Exercise 23.1</a></li>
<li class="chapter" data-level="A.6.2" data-path="appendexA.html"><a href="appendexA.html#exercise-23.2"><i class="fa fa-check"></i><b>A.6.2</b> Exercise 23.2</a></li>
<li class="chapter" data-level="A.6.3" data-path="appendexA.html"><a href="appendexA.html#exercise-23.3"><i class="fa fa-check"></i><b>A.6.3</b> Exercise 23.3</a></li>
<li class="chapter" data-level="A.6.4" data-path="appendexA.html"><a href="appendexA.html#exercise-23.4"><i class="fa fa-check"></i><b>A.6.4</b> Exercise 23.4</a></li>
<li class="chapter" data-level="A.6.5" data-path="appendexA.html"><a href="appendexA.html#exercise-23.5"><i class="fa fa-check"></i><b>A.6.5</b> Exercise 23.5</a></li>
</ul></li>
<li class="chapter" data-level="A.7" data-path="appendexA.html"><a href="appendexA.html#chapter-28"><i class="fa fa-check"></i><b>A.7</b> Chapter 28</a>
<ul>
<li class="chapter" data-level="A.7.1" data-path="appendexA.html"><a href="appendexA.html#exercise-28.1"><i class="fa fa-check"></i><b>A.7.1</b> Exercise 28.1</a></li>
<li class="chapter" data-level="A.7.2" data-path="appendexA.html"><a href="appendexA.html#exercise-28.2"><i class="fa fa-check"></i><b>A.7.2</b> Exercise 28.2</a></li>
<li class="chapter" data-level="A.7.3" data-path="appendexA.html"><a href="appendexA.html#exercise-28.3"><i class="fa fa-check"></i><b>A.7.3</b> Exercise 28.3</a></li>
<li class="chapter" data-level="A.7.4" data-path="appendexA.html"><a href="appendexA.html#exercise-28.4"><i class="fa fa-check"></i><b>A.7.4</b> Exercise 28.4</a></li>
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<li class="chapter" data-level="A.8" data-path="appendexA.html"><a href="appendexA.html#chapter-31"><i class="fa fa-check"></i><b>A.8</b> Chapter 31</a>
<ul>
<li class="chapter" data-level="A.8.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.1"><i class="fa fa-check"></i><b>A.8.1</b> Exercise 31.1</a></li>
<li class="chapter" data-level="A.8.2" data-path="appendexA.html"><a href="appendexA.html#exercise-31.2"><i class="fa fa-check"></i><b>A.8.2</b> Exercise 31.2</a></li>
<li class="chapter" data-level="A.8.3" data-path="appendexA.html"><a href="appendexA.html#exercise-31.3"><i class="fa fa-check"></i><b>A.8.3</b> Exercise 31.3</a></li>
<li class="chapter" data-level="A.8.4" data-path="appendexA.html"><a href="appendexA.html#exercise-31.4"><i class="fa fa-check"></i><b>A.8.4</b> Exercise 31.4</a></li>
<li class="chapter" data-level="A.8.5" data-path="appendexA.html"><a href="appendexA.html#exercise-31.5"><i class="fa fa-check"></i><b>A.8.5</b> Exercise 31.5</a></li>
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<li class="chapter" data-level="A.9" data-path="appendexA.html"><a href="appendexA.html#chapter-34"><i class="fa fa-check"></i><b>A.9</b> Chapter 34</a>
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<li class="chapter" data-level="A.9.1" data-path="appendexA.html"><a href="appendexA.html#exercise-34.1"><i class="fa fa-check"></i><b>A.9.1</b> Exercise 34.1</a></li>
<li class="chapter" data-level="A.9.2" data-path="appendexA.html"><a href="appendexA.html#exercise-34.2"><i class="fa fa-check"></i><b>A.9.2</b> Exercise 34.2</a></li>
<li class="chapter" data-level="A.9.3" data-path="appendexA.html"><a href="appendexA.html#exercise-34.3"><i class="fa fa-check"></i><b>A.9.3</b> Exercise 34.3</a></li>
<li class="chapter" data-level="A.9.4" data-path="appendexA.html"><a href="appendexA.html#exercise-34.4"><i class="fa fa-check"></i><b>A.9.4</b> Exercise 34.4</a></li>
<li class="chapter" data-level="A.9.5" data-path="appendexA.html"><a href="appendexA.html#exercise-33.5"><i class="fa fa-check"></i><b>A.9.5</b> Exercise 33.5</a></li>
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<li class="chapter" data-level="B" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html"><i class="fa fa-check"></i><b>B</b> Uncertainty derivation</a>
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<li class="chapter" data-level="B.1" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-addition-and-subtraction"><i class="fa fa-check"></i><b>B.1</b> Propagation of error for addition and subtraction</a></li>
<li class="chapter" data-level="B.2" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-multiplication-and-division"><i class="fa fa-check"></i><b>B.2</b> Propagation of error for multiplication and division</a></li>
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<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</a>
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<div id="Chapter_22" class="section level1 hasAnchor" number="22">
<h1><span class="header-section-number">Chapter 22</span> The t-test<a href="Chapter_22.html#Chapter_22" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>A t-test is a simple and widely used statistical hypothesis test that relies on the t-distribution introduced in <a href="Chapter_19.html#Chapter_19">Chapter 19</a>.
In this chapter, we will look at three types of t-tests: (1) the one sample t-test, (2) the independent samples t-test, and (3) the paired samples t-test.
We will also look at non-parametric alternatives to t-tests (Wilcoxon and Mann-Whitney tests), which become relevant when the assumptions of t-tests are violated.
The use of all of these tests in jamovi will be demonstrated in <a href="Chapter_23.html#Chapter_23">Chapter 23</a>.</p>
<div id="one-sample-t-test" class="section level2 hasAnchor" number="22.1">
<h2><span class="header-section-number">22.1</span> One sample t-test<a href="Chapter_22.html#one-sample-t-test" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Suppose that a biology teacher has created a new approach to teaching and wants to test whether or not their new approach results in student test scores that are higher than the reported national average of 60.
This teacher should first define their null and alternative hypotheses.</p>
<ul>
<li><span class="math inline">\(H_{0}\)</span>: The mean of student test scores equals 60</li>
<li><span class="math inline">\(H_{A}\)</span>: The mean of student test scores is greater than 60</li>
</ul>
<p>Note that this is a one-sided hypothesis.
The teacher is not interested in whether or not the mean test score of their students is below 60.
They just want to find out if the mean test scores are greater than 60.
Suppose the teacher has 10 students with the following test scores (out of 100).</p>
<pre><code>49.3, 62.9, 73.7, 65.5, 69.6, 70.7, 61.5, 73.4, 61.1, 78.1</code></pre>
<p>The teacher can use a one sample t-test to test <span class="math inline">\(H_{0}\)</span>.
The one sample t-test will test whether the sample mean of test scores (<span class="math inline">\(\bar{y} = 66.58\)</span>) is significantly greater than the reported national average, <span class="math inline">\(\mu_{0} = 60\)</span>.
How does this work?
Recall from <a href="Chapter_16.html#Chapter_16">Chapter 16</a> that, due to the central limit theorem, the distribution of sample means (<span class="math inline">\(\bar{y}\)</span>) will be normally distributed around the true mean <span class="math inline">\(\mu\)</span> as sample size <span class="math inline">\(N\)</span> increases.
At low <span class="math inline">\(N\)</span>, when we need to estimate the true standard deviation (<span class="math inline">\(\sigma\)</span>) from the sample standard deviation (<span class="math inline">\(s\)</span>), we need to correct for a bias and use the t-distribution (see <a href="Chapter_19.html#Chapter_19">Chapter 19</a>).
The logic here is to use the t-distribution as the null distribution for <span class="math inline">\(\bar{y}\)</span>.
If we subtract <span class="math inline">\(\mu_{0}\)</span> from <span class="math inline">\(\bar{y}\)</span>, then we can centre the mean of the null distribution at 0.
We can then divide by the standard error of test scores so that we can compare the deviation of <span class="math inline">\(\bar{y}\)</span> from <span class="math inline">\(\mu_{0}\)</span> in terms of the t-distribution.
This is the same idea as calculating a z-score from <a href="Chapter_16.html#probability-and-z-scores">Section 16.2</a>.
In fact, the equations look almost the same,</p>
<p><span class="math display">\[t_{\bar{y}} = \frac{\bar{y} - \mu_{0}}{\mathrm{SE}(\bar{y})}.\]</span></p>
<p>In the above equation, <span class="math inline">\(\mathrm{SE}(\bar{y})\)</span> is the standard error of <span class="math inline">\(\bar{y}\)</span>.</p>
<p>If the sample mean of test scores is really the same as the population mean <span class="math inline">\(\mu_{0} = 60\)</span>, then <span class="math inline">\(\bar{y}\)</span> should have a t-distribution.
Consequently, values of <span class="math inline">\(t_{\bar{y}}\)</span> far from zero would suggest that the sample mean is improbable given the null distribution predicted if <span class="math inline">\(H_{0}: \mu_{0} = \bar{y}\)</span> is true.
We can calculate <span class="math inline">\(t_{\bar{y}}\)</span> for our above sample (note, <span class="math inline">\(\mathrm{SE}(\bar{y}) = s/\sqrt{N} = 8.334373 / \sqrt{10} = 2.63556\)</span>),</p>
<p><span class="math display">\[t_{\bar{y}} = \frac{66.58 - 60}{2.63556} = 2.496623.\]</span></p>
<p>Our t-statistic is therefore 2.496623 (note that a t-statistic can also be negative; this would just mean that our sample mean is less than <span class="math inline">\(\mu_{0}\)</span>, instead of greater than <span class="math inline">\(\mu_{0}\)</span>, but nothing about the t-test changes if this is the case).
We can see where this value falls on the t-distribution with 9 degrees of freedom in Figure 22.1.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-82"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-82-1.png" alt="A bell curve of the t-distribution is shown with an arrow pointing downward on the x-axis value about halfway between 2 and 3." width="672" />
<p class="caption">
Figure 22.1: A t-distribution is shown with a calculated t-statistic of 2.49556 indicated with a downward arrow.
</p>
</div>
<p>The t-distribution in Figure 22.1 is the probability distribution if <span class="math inline">\(H_{0}\)</span> is true (i.e., the student test scores were sampled from a distribution with a mean of <span class="math inline">\(\mu_{0} = 60\)</span>).
The arrow pointing to the calculated <span class="math inline">\(t_{\bar{y}} = 2.496623\)</span> indicates that if <span class="math inline">\(H_{0}\)</span> is true, then the sample mean of student test scores <span class="math inline">\(\bar{y} = 66.58\)</span> would be very unlikely.
This is because only a small proportion of the probability distribution in Figure 22.1 is greater than or equal to our t-statistic, <span class="math inline">\(t_{\bar{y}} = 2.496623\)</span>.
In fact, the proportion of t-statistics greater than 2.496623 is only about <span class="math inline">\(P =\)</span> 0.017.
Hence, if our null hypothesis is true, then the probability of getting a mean student test score of 66.58 or higher is <span class="math inline">\(P =\)</span> 0.017 (this is our p-value).
It is important to understand the relationship between the t-statistic and the p-value; an interactive application<a href="#fn45" class="footnote-ref" id="fnref45"><sup>45</sup></a> can help visualise.
Typically, we set a threshold level of <span class="math inline">\(\alpha = 0.05\)</span>, below which we conclude that our p-value is statistically significant (see <a href="Chapter_21.html#Chapter_21">Chapter 21</a>).
Consequently, because our p-value is less than 0.05, we reject our null hypothesis and conclude that student test scores are higher than the reported national average.</p>
</div>
<div id="independent-samples-t-test" class="section level2 hasAnchor" number="22.2">
<h2><span class="header-section-number">22.2</span> Independent samples t-test<a href="Chapter_22.html#independent-samples-t-test" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Perhaps the biology teacher is not actually interested in comparing their students’ test results with those of the reported national average.
After all, there might be many reasons their students score differently from the national average that have nothing to do with their new approach to teaching.
To see if their new approach is working, the teacher might instead decide that a better hypothesis to test is whether or not the mean test score from the current year is higher than the mean test score from the class that they taught in the previous year.
We can use <span class="math inline">\(\bar{y}_{1}\)</span> to denote the mean of test scores from the current year, and <span class="math inline">\(\bar{y}_{2}\)</span> to denote the mean of test scores from the previous year.
The test scores of the current year (<span class="math inline">\(y_{1}\)</span>) therefore remain the same as in the example of the one sample t-test from the previous section.</p>
<pre><code>49.3, 62.9, 73.7, 65.5, 69.6, 70.7, 61.5, 73.4, 61.1, 78.1</code></pre>
<p>Suppose that in the previous year, there were 9 students in the class (i.e., one fewer than the current year).
These 9 students received the following test scores (<span class="math inline">\(y_{2}\)</span>).</p>
<pre><code>57.4, 52.4, 70.5, 71.6, 46.1, 60.4, 70.0, 64.5, 58.8</code></pre>
<p>The mean score from last year was <span class="math inline">\(\bar{y}_{2} = 61.30\)</span>, which does appear to be lower than the mean score of the current year, <span class="math inline">\(\bar{y}_{1} = 66.58\)</span>.
But is the difference between these two means statistically significant?
In other words, were the test scores from each year sampled from a population with the same mean, such that the population mean of the previous year (<span class="math inline">\(\mu_{2}\)</span>) and the current year (<span class="math inline">\(\mu_{1}\)</span>) are the same?
This is the null hypothesis, <span class="math inline">\(H_{0}: \mu_{1} = \mu_{2}\)</span>.</p>
<p>The general idea for testing this null hypothesis is the same as it was in the one sample t-test.
In both cases, we want to calculate a t-statistic, then see where it falls along the t-distribution to decide whether or not to reject <span class="math inline">\(H_{0}\)</span>.
In this case, our t-statistic (<span class="math inline">\(t_{\bar{y}_{1} - \bar{y}_{2}}\)</span>) is calculated slightly differently,</p>
<p><span class="math display">\[t_{\bar{y}_{1} - \bar{y}_{2}} = \frac{\bar{y}_{1} - \bar{y}_{2}}{\mathrm{SE}(\bar{y})}\]</span></p>
<p>The logic is the same as the one sample t-test.
If <span class="math inline">\(\mu_{1} = \mu_{2}\)</span>, then we also would expect <span class="math inline">\(\bar{y}_{1} = \bar{y}_{2}\)</span> (i.e., <span class="math inline">\(\bar{y}_{1} - \bar{y}_{2} = 0\)</span>).
Differences between <span class="math inline">\(\bar{y}_{1}\)</span> and <span class="math inline">\(\bar{y}_{2}\)</span> cause the t-statistic to be either above or below 0, and we can map this deviation of <span class="math inline">\(t_{\bar{y}_{1} - \bar{y}_{2}}\)</span> from 0 to the probability density of the t-distribution after standardising by the standard error (<span class="math inline">\(\mathrm{SE}(\bar{y})\)</span>).</p>
<p>What is <span class="math inline">\(\mathrm{SE}(\bar{y})\)</span> in this case?
After all, there are two different samples <span class="math inline">\(y_{1}\)</span> and <span class="math inline">\(y_{2}\)</span>, so could the two samples not have <em>different</em> standard errors?
This could indeed be the case, and how we actually conduct the independent samples t-test depends on whether or not we are willing to assume that the two samples came from populations with the same variance (i.e., <span class="math inline">\(\sigma_{1} = \sigma_{2}\)</span>).
If we are willing to make this assumption, then we can pool the variances (<span class="math inline">\(s^{2}_{p}\)</span>) together to get a combined (more accurate) estimate of the standard error <span class="math inline">\(\mathrm{SE}(\bar{y})\)</span> from both samples<a href="#fn46" class="footnote-ref" id="fnref46"><sup>46</sup></a><span class="math inline">\(^{,}\)</span><a href="#fn47" class="footnote-ref" id="fnref47"><sup>47</sup></a>.
This version of the independent samples t-test is called the ‘Student’s t-test’.</p>
<p>If we are unwilling to assume that <span class="math inline">\(y_{1}\)</span> and <span class="math inline">\(y_{2}\)</span> have the same variance, then we need to use an alternative version of the independent samples t-test.
This alternative version is called the Welch’s t-test <span class="citation">(<a href="#ref-Welch1938" role="doc-biblioref">Welch, 1938</a>)</span>, also known as the unequal variances t-test <span class="citation">(<a href="#ref-Dytham2011" role="doc-biblioref">Dytham, 2011</a>; <a href="#ref-Ruxton2006" role="doc-biblioref">Ruxton, 2006</a>)</span>.
In contrast to the Student’s t-test, the Welch’s t-test does not pool the variances of the samples together<a href="#fn48" class="footnote-ref" id="fnref48"><sup>48</sup></a>.
While there are some mathematical differences between the Student’s and Welch’s independent samples t-tests, the general concept is the same.</p>
<p>This raises the question, when is it acceptable to assume that <span class="math inline">\(y_{1}\)</span> and <span class="math inline">\(y_{2}\)</span> have the same variance?
The sample variance of <span class="math inline">\(s^{2}_{1} = 69.46\)</span> and <span class="math inline">\(s^{2}_{2} = 76.15\)</span>.
Is this close enough to treat them as the same?
Like a lot of choices in statistics, there is no clear right or wrong answer.
In theory, if both samples do come from a population with the same variance (<span class="math inline">\(\sigma^{2}_{1} = \sigma^{2}_{2}\)</span>), then the pooled variance is better because it gives us a bit more statistical power; we can correctly reject the null hypothesis more often when it is actually false (i.e., it decreases the probability of a Type II error).
Nevertheless, the increase in statistical power is quite low, and the risk of pooling the variances when they actually are not the same increases the risk that we reject the null hypothesis when it is actually true (i.e., it increases the probability of a Type I error, which we definitely do not want!).
For this reason, some researchers advocate using the Welch’s t-test by default, unless there is a very good reason to believe <span class="math inline">\(y_{1}\)</span> and <span class="math inline">\(y_{2}\)</span> are sampled from populations with the same variance <span class="citation">(<a href="#ref-Delacre2017" role="doc-biblioref">Delacre et al., 2017</a>; <a href="#ref-Ruxton2006" role="doc-biblioref">Ruxton, 2006</a>)</span>.</p>
<p>Here we will adopt the traditional approach of first testing the null hypothesis that <span class="math inline">\(\sigma^{2}_{1} = \sigma^{2}_{2}\)</span> using a homogeneity of variances test.
If we fail to reject this null hypothesis (i.e., <span class="math inline">\(P > 0.05\)</span>), then we will use the Student’s t-test, and if we reject it (i.e., <span class="math inline">\(P < 0.05\)</span>), then we will use the Welch’s t-test.
This approach is mostly used for pedagogical reasons; in practice, defaulting to the Welch’s t-test is fine <span class="citation">(<a href="#ref-Delacre2017" role="doc-biblioref">Delacre et al., 2017</a>; <a href="#ref-Ruxton2006" role="doc-biblioref">Ruxton, 2006</a>)</span>.
Testing for homogeneity of variances is quite straightforward in most statistical programs, and we will save the conceptual and mathematical details of this for when we look at the F-distribution in <a href="Chapter_24.html#Chapter_24">Chapter 24</a>.
But the general idea is that if <span class="math inline">\(\sigma^{2}_{1} = \sigma^{2}_{2}\)</span>, then the ratio of variances (<span class="math inline">\(\sigma^{2}_{1}/\sigma^{2}_{2}\)</span>) has its own null distribution (like the normal distribution, or the t-distribution), and we can see the probability of getting a deviation of <span class="math inline">\(\sigma^{2}_{1}/\sigma^{2}_{2}\)</span> from 1 if <span class="math inline">\(\sigma^{2}_{1} = \sigma^{2}_{2}\)</span> is true.</p>
<p>In the case of the test scores from the two samples of students (<span class="math inline">\(y_{1}\)</span> and <span class="math inline">\(y_{2}\)</span>), a homogeneity of variance test reveals no evidence that <span class="math inline">\(s^{2}_{1} = 69.46\)</span> and <span class="math inline">\(s^{2}_{2} = 76.15\)</span> are significantly different (<span class="math inline">\(P = 0.834\)</span>).
We can therefore use the pooled variance and the Student’s independent samples t-test.
We can calculate <span class="math inline">\(\mathrm{SE}(\bar{y}) = 3.915144\)</span> using the formula for <span class="math inline">\(s_{p}\)</span> (again, this is not something that ever actually needs to be done by hand), then find <span class="math inline">\(t_{\bar{y}_{1} - \bar{y}_{2}}\)</span>,</p>
<p><span class="math display">\[t_{\bar{y}_{1} - \bar{y}_{2}} = \frac{\bar{y}_{1} - \bar{y}_{2}}{\mathrm{SE}(\bar{y})} = \frac{66.58 - 61.3}{3.915144} = 1.348609.\]</span></p>
<p>As with the one-sample t-test, we can identify the position of <span class="math inline">\(t_{\bar{y}_{1} - \bar{y}_{2}}\)</span> on the t-distribution (Figure 22.2).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-83"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-83-1.png" alt="A bell curve of the t-distribution is shown with an arrow pointing downward on the x-axis value around 1.3." width="672" />
<p class="caption">
Figure 22.2: A t-distribution is shown with a calculated t-statistic of 1.348609 indicated with a downward arrow.
</p>
</div>
<p>The proportion of t-scores that are higher than <span class="math inline">\(t_{\bar{y}_{1} - \bar{y}_{2}} = 1.348609\)</span> is about 0.098.
In other words, given that the null hypothesis is true, the probability of getting a t-statistic this high is <span class="math inline">\(P = 0.098\)</span>.
Because this p-value exceeds our critical value of <span class="math inline">\(\alpha = 0.05\)</span>, we do not reject the null hypothesis.
We therefore should conclude that the mean of test scores from the current year (<span class="math inline">\(\bar{y}_{1}\)</span>) is not significantly different from the mean of test scores in the previous year (<span class="math inline">\(\bar{y}_{2}\)</span>).
The biology teacher in our example might therefore conclude that mean test results have not improved from the previous year.</p>
</div>
<div id="paired-samples-t-test" class="section level2 hasAnchor" number="22.3">
<h2><span class="header-section-number">22.3</span> Paired samples t-test<a href="Chapter_22.html#paired-samples-t-test" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>There is one more type of t-test to consider.
The paired samples t-test is applied when the data points in one sample can be naturally paired with those in another sample.
In this case, data points between samples are not independent.
For example, we can consider the student test scores (<span class="math inline">\(y_{1}\)</span>) yet again.</p>
<pre><code>49.3, 62.9, 73.7, 65.5, 69.6, 70.7, 61.5, 73.4, 61.1, 78.1</code></pre>
<p>Suppose that the teacher gave these same 10 students (S1–S10) a second test and wanted to see if the mean student score changed from one test to the next (i.e., a two-sided hypothesis).</p>
<table style="width:100%;">
<caption><strong>TABLE 22.1</strong> Test scores from 10 students (S1–S10) for two different tests in a hypothetical biology education example.</caption>
<colgroup>
<col width="15%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">S1</th>
<th align="center">S2</th>
<th align="center">S3</th>
<th align="center">S4</th>
<th align="center">S5</th>
<th align="center">S6</th>
<th align="center">S7</th>
<th align="center">S8</th>
<th align="center">S9</th>
<th align="center">S10</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Test 1</strong></td>
<td align="center">49.3</td>
<td align="center">62.9</td>
<td align="center">73.7</td>
<td align="center">65.5</td>
<td align="center">69.6</td>
<td align="center">70.7</td>
<td align="center">61.5</td>
<td align="center">73.4</td>
<td align="center">61.1</td>
<td align="center">78.1</td>
</tr>
<tr class="even">
<td align="center"><strong>Test 2</strong></td>
<td align="center">46.6</td>
<td align="center">62.7</td>
<td align="center">73.8</td>
<td align="center">58.3</td>
<td align="center">66.8</td>
<td align="center">69.7</td>
<td align="center">64.5</td>
<td align="center">71.3</td>
<td align="center">64.5</td>
<td align="center">78.8</td>
</tr>
<tr class="odd">
<td align="center"><strong>Change</strong></td>
<td align="center">-2.7</td>
<td align="center">-0.2</td>
<td align="center">0.1</td>
<td align="center">-7.2</td>
<td align="center">-2.8</td>
<td align="center">-1</td>
<td align="center">3</td>
<td align="center">-2.1</td>
<td align="center">3.4</td>
<td align="center">0.7</td>
</tr>
</tbody>
</table>
<p>In this case, what we are really interested in is the <em>change</em> in scores from Test 1 to Test 2.
We want to test the null hypothesis that this change is zero.
This is actually the same test as the one-sample t-test.
We are just substituting the mean difference in values (i.e., ‘Change’ in Table 22.1) for <span class="math inline">\(\bar{y}\)</span> and setting <span class="math inline">\(\mu_{0} = 0\)</span>.
We can calculate <span class="math inline">\(\bar{y} = -0.88\)</span> and <span class="math inline">\(\mathrm{SE}(\bar{y})=0.9760237\)</span>, then set up the t-test as before,</p>
<p><span class="math display">\[t_{\bar{y}} = \frac{-0.88 - 0}{0.9760237} = -0.9016175.\]</span></p>
<p>Again, we can find the location of our t-statistic <span class="math inline">\(t_{\bar{y}} = -0.9016175\)</span> on the t-distribution (Figure 22.3).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-85"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-85-1.png" alt="A bell curve of the t-distribution is shown with an arrow pointing downward on the x-axis value around -0.9." width="672" />
<p class="caption">
Figure 22.3: A t-distribution is shown with a calculated t-statistic of <span class="math inline">\(-0.9016175\)</span> indicated with a downward arrow.
</p>
</div>
<p>Since this is a two-sided hypothesis, we want to know the probability of getting a t-statistic as extreme as <span class="math inline">\(-0.9016175\)</span> (i.e., either <span class="math inline">\(\pm 0.9016175\)</span>) given that the null distribution is true.
In the above t-distribution, 95% of the probability density lies between <span class="math inline">\(t = -2.26\)</span> and <span class="math inline">\(t = 2.26\)</span>.
Consequently, our calculated <span class="math inline">\(t_{\bar{y}} = -0.9016175\)</span> is not sufficiently extreme to reject the null hypothesis.
The p-value associated with <span class="math inline">\(t_{\bar{y}} = -0.9016175\)</span> is <span class="math inline">\(P = 0.391\)</span>.
We therefore fail to reject <span class="math inline">\(H_{0}\)</span> and conclude that there is no significant difference in student test scores from Test 1 to Test 2.</p>
</div>
<div id="assumptions-of-t-tests" class="section level2 hasAnchor" number="22.4">
<h2><span class="header-section-number">22.4</span> Assumptions of t-tests<a href="Chapter_22.html#assumptions-of-t-tests" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>We make some potentially important assumptions when using t-tests.
A consequence of violating these assumptions is a misleading Type I error rate.
That is, if our data do not fit the assumptions of our statistical test, then we might not actually be rejecting our null hypothesis at the <span class="math inline">\(\alpha = 0.05\)</span> level.
We might unknowingly be rejecting <span class="math inline">\(H_{0}\)</span> when it is true at a much higher <span class="math inline">\(\alpha\)</span> value, and therefore concluding that we have evidence supporting the alternative hypothesis <span class="math inline">\(H_{A}\)</span> when we really do not.
It is important to recognise the assumptions that we are making when using any statistical test (including t-tests).
If our assumptions are violated, we might need to use a different test, or perhaps apply some kind of transformation on the data.
Assumptions that we make when conducting a t-test are as follows:</p>
<ul>
<li>Data are continuous (i.e., not count or categorical data)</li>
<li>Sample observations are a random sample from the population</li>
<li>Sample means are normally distributed around the true mean</li>
</ul>
<p>Note that if we are running a Student’s independent samples t-test that pools sample variances (rather than a Welch’s t-test), then we are also assuming that sample variances are the same (i.e., homogeneity of variance).
The last bullet point concerning normally distributed sample means is frequently misunderstood to mean that the sample data themselves need to be normally distributed.
This is not the case <span class="citation">(<a href="#ref-Johnson1995" role="doc-biblioref">Johnson, 1995</a>; <a href="#ref-Lumley2002" role="doc-biblioref">Lumley et al., 2002</a>)</span>.
Instead, what we are really concerned with is the distribution of sample means (<span class="math inline">\(\bar{y}\)</span>) around the true mean (<span class="math inline">\(\mu\)</span>).
And given a sufficiently large sample size, a normal distribution is assured due to the central limit theorem (see <a href="Chapter_16.html#Chapter_16">Chapter 16</a>).</p>
<p>Moreover, while a normally distributed variable is not <em>necessary</em> for satisfying the assumptions of a t-test (or many other tests introduced in this book), it is <em>sufficient</em>.
In other words, if the variable being measured is normally distributed, then the sample means will also be normally distributed around the true mean (even for a low sample size).
So when is a sample size large enough, or close enough to being normally distributed, for the assumption of normality to be satisfied?
There really is not a definitive answer to this question, and the truth is that most statisticians will prefer to use a histogram (or some other visualisation approach) and their best judgement to decide if the assumption of normality is likely to be violated.</p>
<p>This book will take the traditional approach of running a statistical test called the Shapiro-Wilk test to test the null hypothesis that data are normally distributed.
If we reject the null hypothesis (when <span class="math inline">\(P < 0.05\)</span>), then we will conclude that the assumption of normality is violated, and the t-test is not appropriate.
The details of how the Shapiro-Wilk test works are not important for now, but the test can be easily run using jamovi <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The jamovi project, 2024</a>)</span>.
If we reject the null hypothesis that the data are normally distributed, then we can use one of two methods to run our statistical test.</p>
<ol style="list-style-type: decimal">
<li>Transform the data in some way (e.g., take the log of all values) to improve normality.</li>
<li>Use a non-parametric alternative test.</li>
</ol>
<p>The word ‘non-parametric’ in this context just means that there are no assumptions (or very few) about the shape of the distribution <span class="citation">(<a href="#ref-Dytham2011" role="doc-biblioref">Dytham, 2011</a>)</span>.
We will consider the non-parametric equivalents of the one-sample t-test (the Wilcoxon test) and independent samples t-test (Mann-Whitney U test) in the next section.
But first, we can show how transformations can be used to improve the fit of the data to satisfy model assumptions.</p>
<p>Often data will have a skewed distribution (see <a href="Chapter_13.html#Chapter_13">Chapter 13</a>).
For example, in Figure 22.4A, we have a dataset (sample size <span class="math inline">\(N = 200\)</span>) with a large positive skew (i.e., it is right-skewed).
Most values are in the same general area, but with some values being especially high.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-86"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-86-1.png" alt="A two paneled figure in which a histogram with a skewed distribution is shown to the left and a histogram with a normal distribution is shown to the right." width="672" />
<p class="caption">
Figure 22.4: Set of values with a high positive skew (A), which, when log-transformed (i.e., when we take the natural log of all values), have a normal distribution (B).
</p>
</div>
<p>Using a t-test on the variable <span class="math inline">\(X\)</span> shown in Figure 22.4A is probably not a good idea.
But taking the natural log of all the values of <span class="math inline">\(X\)</span> makes the dataset more normally distributed (Figure 22.4B), thereby more convincingly satisfying the normality assumption required by the t-test.
This might seem a bit suspicious at first.
Is it really okay to just take the logarithm of all the data instead of the actual values that were measured?
Actually, there is no real scientific or statistical reason that we need to use the original scale <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
Using the log or the square-root of a set of numbers is perfectly fine if it helps satisfy the assumptions of a statistical test.</p>
</div>
<div id="non-parametric-alternatives" class="section level2 hasAnchor" number="22.5">
<h2><span class="header-section-number">22.5</span> Non-parametric alternatives<a href="Chapter_22.html#non-parametric-alternatives" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>If we find that the assumption of normality is not satisfied, and a transformation of the data cannot help, then we can consider using non-parametric alternatives to a t-test.
These alternatives include the Wilcoxon test and the Mann-Whitney U test.</p>
<div id="wilcoxon-test" class="section level3 hasAnchor" number="22.5.1">
<h3><span class="header-section-number">22.5.1</span> Wilcoxon test<a href="Chapter_22.html#wilcoxon-test" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>The Wilcoxon test (also called the ‘Wilcoxon signed-rank test’) is the non-parametric alternative to a one sample t-test (or a paired t-test).
Instead of using the actual data, the Wilcoxon test ranks all of the values in the dataset, then sums up their signs (either positive or negative).
The general idea is that we can compare the sum of the <em>ranks</em> of the actual data with what would be predicted by the null hypothesis.
It tests the null hypothesis that the <em>median</em> (<span class="math inline">\(M\)</span>) is significantly different from some given value<a href="#fn49" class="footnote-ref" id="fnref49"><sup>49</sup></a>.
An example will make it easier to see how it works.
We can use the same hypothetical dataset on student test scores.</p>
<pre><code>49.3, 62.9, 73.7, 65.5, 69.6, 70.7, 61.5, 73.4, 61.1, 78.1</code></pre>
<p>The first step is to subtract the null hypothesis value (<span class="math inline">\(M = 60\)</span>, if we again set <span class="math inline">\(H_{0}\)</span> to be that the average student test score equals 60) from each value (<span class="math inline">\(49.3 - 60 = -10.7\)</span>, <span class="math inline">\(62.9 - 60 = 2.9\)</span>, etc.).</p>
<pre><code>-10.7, 2.9, 13.7, 5.5, 9.6, 10.7, 1.5, 13.4, 1.1, 18.1</code></pre>
<p>We need to note the sign of each value as negative (<span class="math inline">\(-\)</span>) or positive (<span class="math inline">\(+\)</span>).</p>
<pre><code>-, +, +, +, +, +, +, +, +, +</code></pre>
<p>Next, we need to compute the absolute values of the numbers (i.e., <span class="math inline">\(|-10.7| = 10.7\)</span>, <span class="math inline">\(|2.9| = 2.9\)</span>, <span class="math inline">\(|13.7| = 13.7\)</span>, etc.).</p>
<pre><code>10.7, 2.9, 13.7, 5.5, 9.6, 10.7, 1.5, 13.4, 1.1, 18.1</code></pre>
<p>We then rank these values from lowest to highest and record the sign of each value.</p>
<pre><code>6.5, 3.0, 9.0, 4.0, 5.0, 6.5, 2.0, 8.0, 1.0, 10.0</code></pre>
<p>Note that both the first and sixth position had the same value (10.7), so instead of ranking them as 6 and 7, we split the difference and rank both as 6.5.
Now, we can calculate the sum of the negative ranks (<span class="math inline">\(W^{-}\)</span>), and the positive ranks <span class="math inline">\(W^{+}\)</span>.
In this case, the negative ranks are easy; there is only one value (the first one), so the sum is just 6.5,</p>
<p><span class="math display">\[W^{-} = 6.5\]</span></p>
<p>The positive ranks are in positions 2–10, and the rank values in these positions are 3, 9, 4, 5, 6.5, 2, 8, 1, and 10.
The sum of our positive ranks is therefore,</p>
<p><span class="math display">\[W^{+} = 3.0 + 9.0 + 4.0 + 5.0 + 6.5 + 2.0 + 8.0 + 1.0 + 10.0 = 48.5\]</span></p>
<p>Note that <span class="math inline">\(W^{-}\)</span> plus <span class="math inline">\(W^{+}\)</span> (i.e., 6.5 + 48.5 = 55 in the example here) will always be the same for a given sample size <span class="math inline">\(N\)</span> (in this case, <span class="math inline">\(N = 10\)</span>),</p>
<p><span class="math display">\[W = \frac{N \left(N + 1 \right)}{2}.\]</span></p>
<p>What the Wilcoxon test is doing is calculating the probability of getting a value of <span class="math inline">\(W^{+}\)</span> as or more extreme than would be the case if the null hypothesis is true.
Note that if, e.g., there are an equal number of values above and below the median, then both <span class="math inline">\(W^{-}\)</span> and <span class="math inline">\(W^{+}\)</span> will be relatively low and about the same value.
This is because the ranks of the values below 0 (which we multiply by <span class="math inline">\(-1\)</span>) and above 0 (which we multiply by 1) will be about the same.
But if there are a lot more values above the median than expected (as with the example above), then <span class="math inline">\(W^{+}\)</span> will be relatively high.
And if there are a lot more values below the median than expected, then <span class="math inline">\(W^{+}\)</span> will be relatively low.</p>
<p>To find the probability of <span class="math inline">\(W^{+}\)</span> being as low or high as it is given that the null hypothesis is true (i.e., the p-value), we need to compare the test statistic <span class="math inline">\(W^{+}\)</span> to its distribution under the null hypothesis (note that we can also conduct a one-tailed hypothesis, in which case we are testing if <span class="math inline">\(W^{+}\)</span> is either higher or lower than expected given <span class="math inline">\(H_{0}\)</span>).
The old way of doing this is to compare the calculated <span class="math inline">\(W^{+}\)</span> to threshold values from a Wilcoxon Signed-Ranks Table.
This critical value table is no longer necessary, and statistical software such as jamovi will calculate a p-value for us.
For the example above, the p-value associated with <span class="math inline">\(W^{+} = 48.5\)</span> and <span class="math inline">\(N = 10\)</span> is <span class="math inline">\(P = 0.037\)</span>.
Since this p-value is less than our threshold of <span class="math inline">\(\alpha = 0.05\)</span>, we can reject the null hypothesis and conclude that the median of our dataset is significantly different from 60.</p>
<p>Note that we can also use a Wilcoxon signed rank test as a non-parametric equivalent to a paired t-test.
In this case, instead of subtracting out the null hypothesis of our median value (e.g., <span class="math inline">\(H_{0}: M = 60\)</span> in the example above), we just need to subtract the paired values.
Consider again the example of the two different tests introduced for the <a href="#paired-sample-t-test">paired samples t-test</a> above (Table 22.2).</p>
<table style="width:100%;">
<caption><strong>TABLE 22.2</strong> Test scores from 10 students (S1–S10) for two different tests in a hypothetical biology education example.</caption>
<colgroup>
<col width="15%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
<col width="8%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">S1</th>
<th align="center">S2</th>
<th align="center">S3</th>
<th align="center">S4</th>
<th align="center">S5</th>
<th align="center">S6</th>
<th align="center">S7</th>
<th align="center">S8</th>
<th align="center">S9</th>
<th align="center">S10</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Test 1</strong></td>
<td align="center">49.3</td>
<td align="center">62.9</td>
<td align="center">73.7</td>
<td align="center">65.5</td>
<td align="center">69.6</td>
<td align="center">70.7</td>
<td align="center">61.5</td>
<td align="center">73.4</td>
<td align="center">61.1</td>
<td align="center">78.1</td>
</tr>
<tr class="even">
<td align="center"><strong>Test 2</strong></td>
<td align="center">46.6</td>
<td align="center">62.7</td>
<td align="center">73.8</td>
<td align="center">58.3</td>
<td align="center">66.8</td>
<td align="center">69.7</td>
<td align="center">64.5</td>
<td align="center">71.3</td>
<td align="center">64.5</td>
<td align="center">78.8</td>
</tr>
<tr class="odd">
<td align="center"><strong>Change</strong></td>
<td align="center">-2.7</td>
<td align="center">-0.2</td>
<td align="center">0.1</td>
<td align="center">-7.2</td>
<td align="center">-2.8</td>
<td align="center">-1</td>
<td align="center">3</td>
<td align="center">-2.1</td>
<td align="center">3.4</td>
<td align="center">0.7</td>
</tr>
</tbody>
</table>
<p>If the ‘Change’ values in Table 22.2 were not normally distributed, then we could apply a Wilcoxon test to test the null hypothesis that the median value of <span class="math inline">\(Test\:2 - Test\:1 = 0\)</span> (note that a Shapiro-Wilk normality test does not reject the null hypothesis that the difference between test scores is normally distributed, so the paired t-test would be preferred in this case).
To do this, we would first note the sign of each value as negative or positive.</p>
<pre><code>-, -, +, -, -, -, +, -, +, +</code></pre>
<p>Next, we would rank the absolute values of the changes.</p>
<pre><code>6, 2, 1, 10, 7, 4, 8, 5, 9, 3</code></pre>
<p>If we then sum the ranks, we get <span class="math inline">\(W^{-} = 6 + 2 + 10 + 7 + 4 + 5 = 34\)</span> and <span class="math inline">\(W^{+} = 1 + 8 + 9 + 3 = 21\)</span>.
The p-value associated with <span class="math inline">\(W^{+} = 21\)</span> and <span class="math inline">\(N = 10\)</span> in a two-tailed test is <span class="math inline">\(P = 0.557\)</span>, so we do not reject the null hypothesis that Test 1 and Test 2 have the same median.</p>
</div>
<div id="mann-whitney-u-test" class="section level3 hasAnchor" number="22.5.2">
<h3><span class="header-section-number">22.5.2</span> Mann-Whitney U test<a href="Chapter_22.html#mann-whitney-u-test" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>A non-parametric alternative to the independent samples t-test is the Mann-Whitney U test.
That is, a Mann-Whitney U test can be used if we want to know whether the median of two independent groups is significantly different.
Like the Wilcoxon test, the Mann-Whitney U test uses the ranks of values rather than the values themselves.
In the Mann-Whitney U test, the general idea is to rank all of the data across both groups, then see if the sum of the ranks is significantly different <span class="citation">(<a href="#ref-Fryer1966" role="doc-biblioref">Fryer, 1966</a>; <a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
To demonstrate this, we can again consider the same hypothetical dataset used when demonstrating the <a href="Chapter_22.html#independent-samples-t-test">independent samples t-test</a> above.
Test scores from the current year (<span class="math inline">\(y_{1}\)</span>) are below.</p>
<pre><code>49.3, 62.9, 73.7, 65.5, 69.6, 70.7, 61.5, 73.4, 61.1, 78.1</code></pre>
<p>We want to know if the median of the above scores is significantly different from the median scores in the previous year (<span class="math inline">\(y_{2}\)</span>) shown below.</p>
<pre><code>57.4, 52.4, 70.5, 71.6, 46.1, 60.4, 70.0, 64.5, 58.8</code></pre>
<p>There are 19 values in total, 10 values for <span class="math inline">\(y_{1}\)</span> and 9 values for <span class="math inline">\(y_{2}\)</span>.
We therefore rank <em>all</em> of the above values from 1 to 19.
For <span class="math inline">\(y_{1}\)</span>, the ranks are below.</p>
<pre><code>2, 9, 18, 11, 12, 15, 8, 17, 7, 19</code></pre>
<p>For <span class="math inline">\(y_{2}\)</span>, the ranks are below.</p>
<pre><code>4, 3, 14, 16, 1, 6, 13, 10, 5</code></pre>
<p>This might be easier to see if we present it as a table showing the test Year (<span class="math inline">\(y_{1}\)</span> versus <span class="math inline">\(y_{2}\)</span>), test Score, and test Rank (Table 22.3).</p>
<table style="width:32%;">
<caption><strong>TABLE 22.3</strong> Test scores from different students across 2 years, and the overall rank of each test score, in a hypothetical biology education example.</caption>
<colgroup>
<col width="9%" />
<col width="11%" />
<col width="11%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Year</th>
<th align="center">Score</th>
<th align="center">Rank</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">1</td>
<td align="center">49.3</td>
<td align="center">2</td>
</tr>
<tr class="even">
<td align="center">1</td>
<td align="center">62.9</td>
<td align="center">9</td>
</tr>
<tr class="odd">
<td align="center">1</td>
<td align="center">73.7</td>
<td align="center">18</td>
</tr>
<tr class="even">
<td align="center">1</td>
<td align="center">65.5</td>
<td align="center">11</td>
</tr>
<tr class="odd">
<td align="center">1</td>
<td align="center">69.6</td>
<td align="center">12</td>
</tr>
<tr class="even">
<td align="center">1</td>
<td align="center">70.7</td>
<td align="center">15</td>
</tr>
<tr class="odd">
<td align="center">1</td>
<td align="center">61.5</td>
<td align="center">8</td>
</tr>
<tr class="even">
<td align="center">1</td>
<td align="center">73.4</td>
<td align="center">17</td>
</tr>
<tr class="odd">
<td align="center">1</td>
<td align="center">61.1</td>
<td align="center">7</td>
</tr>
<tr class="even">
<td align="center">1</td>
<td align="center">78.1</td>
<td align="center">19</td>
</tr>
<tr class="odd">
<td align="center">2</td>
<td align="center">57.4</td>
<td align="center">4</td>
</tr>
<tr class="even">
<td align="center">2</td>
<td align="center">52.4</td>
<td align="center">3</td>
</tr>
<tr class="odd">
<td align="center">2</td>
<td align="center">70.5</td>
<td align="center">14</td>
</tr>
<tr class="even">
<td align="center">2</td>
<td align="center">71.6</td>
<td align="center">16</td>
</tr>
<tr class="odd">
<td align="center">2</td>
<td align="center">46.1</td>
<td align="center">1</td>
</tr>
<tr class="even">
<td align="center">2</td>
<td align="center">60.4</td>
<td align="center">6</td>
</tr>
<tr class="odd">
<td align="center">2</td>
<td align="center">70</td>
<td align="center">13</td>
</tr>
<tr class="even">
<td align="center">2</td>
<td align="center">64.5</td>
<td align="center">10</td>
</tr>
<tr class="odd">
<td align="center">2</td>
<td align="center">58.8</td>
<td align="center">5</td>
</tr>
</tbody>
</table>
<p>What we need to do now is sum the ranks for <span class="math inline">\(y_{1}\)</span> and <span class="math inline">\(y_{2}\)</span>.
If we add up the <span class="math inline">\(y_{1}\)</span> ranks, then we get a value of <span class="math inline">\(R_{1}=\)</span> 118.
If we add up the <span class="math inline">\(y_{2}\)</span> ranks, then we get a value of <span class="math inline">\(R_{2}=\)</span> 72.
We can then calculate a value <span class="math inline">\(U_{1}\)</span> from <span class="math inline">\(R_{1}\)</span> and the sample size of <span class="math inline">\(y_{1}\)</span> (<span class="math inline">\(N_{1}\)</span>),</p>
<p><span class="math display">\[U_{1} = R_{1} - \frac{N_{1}\left(N_{1} + 1 \right)}{2}.\]</span></p>