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<title>Chapter 31 Practical. Analysis of counts and correlations | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
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<meta name="twitter:title" content="Chapter 31 Practical. Analysis of counts and correlations | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
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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>
</ul></li>
<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>
<ul>
<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>
</ul></li>
<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>
</ul></li>
<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>
</ul></li>
<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>
</ul></li>
<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>
<ul>
<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_31" class="section level1 hasAnchor" number="31">
<h1><span class="header-section-number">Chapter 31</span> <em>Practical</em>. Analysis of counts and correlations<a href="Chapter_31.html#Chapter_31" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>This chapter focuses on applying the concepts from <a href="Chapter_29.html#Chapter_29">Chapter 29</a> and <a href="Chapter_30.html#Chapter_30">Chapter 30</a> in jamovi <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The jamovi project, 2024</a>)</span>.
Exercises in this chapter will use the <a href="#chi-squared-goodness-of-fit">Chi-square goodness of fit</a> test, the <a href="#chi-squared-test-of-association">Chi-square test of association</a>,
and the <a href="Chapter_30.html#correlation-hypothesis-testing">correlation coefficient</a>.
For all of these examples, this chapter will use a dataset inspired by <span class="citation">Burrows et al. (<a href="#ref-Burrows2022" role="doc-biblioref">2022</a>)</span>.
This experimental work tested the effects of radiation on bumblebee nectar consumption, carbon dioxide output, and body mass in different bee colonies.</p>
<p>The chapter will use the bumblebee dataset<a href="#fn74" class="footnote-ref" id="fnref74"><sup>74</sup></a>.
This dataset includes variables for the radiation level experienced by the bee (radiation), the colony from which the bee came (colony), whether or not the bee survived to the end of the 30-day experiment (survived), the mass of the bee in grams at the beginning of the experiment (mass), the output of carbon dioxide put out by the bee (CO<sub>2</sub>) in micromoles per minute, and the daily volume of nectar consumed by the bee in millilitres (nectar).</p>
<div id="survival-goodness-of-fit" class="section level2 hasAnchor" number="31.1">
<h2><span class="header-section-number">31.1</span> Survival goodness of fit<a href="Chapter_31.html#survival-goodness-of-fit" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Suppose that we want to run a simple goodness of fit test to determine whether or not bees are equally likely to survive versus die in the experiment.
If this is the case, then we would expect to see the same number of living and dead bees in the dataset.
We can use a Chi-square goodness of fit test to answer this question.
What are the null and alternative hypotheses for this <span class="math inline">\(\chi^{2}\)</span> goodness of fit test?</p>
<ul>
<li><p><span class="math inline">\(H_{0}\)</span>: _________________</p></li>
<li><p><span class="math inline">\(H_{A}\)</span>: _________________</p></li>
</ul>
<p>What is the sample size (<span class="math inline">\(N\)</span>) of the dataset?</p>
<p><span class="math inline">\(N =\)</span> __________________</p>
<p>Based on this sample size, what are the expected counts for bees that survived and died?</p>
<p>Survived (<span class="math inline">\(E_{\mathrm{surv}}\)</span>): _________________</p>
<p>Died (<span class="math inline">\(E_{\mathrm{died}}\)</span>): ______________</p>
<p>Next, we can find the observed counts of bees that survived and died.
To do this, we need to use the Frequency tables option in jamovi.
We did this once in <a href="Chapter_17.html#Chapter_17">Chapter 17</a> for calculating probabilities.
As a reminder, to find the counts of bumblebees that survived (Yes) or did not survive (No), we need to go to the Exploration toolbar in jamovi, then choose ‘Descriptives’.
Place ‘Survival’ in the Variables box, then check the box for ‘Frequency tables’ below.
A Frequencies table will appear in the panel on the right.
Write down the observed counts of bees that survived and died.</p>
<p>Survived (<span class="math inline">\(O_{\mathrm{surv}}\)</span>): _________________</p>
<p>Died (<span class="math inline">\(O_{\mathrm{died}}\)</span>): ______________</p>
<p>Try to use the formula in <a href="#chi-squared-goodness-of-fit">Section 29.2</a> to calculate the <span class="math inline">\(\chi^{2}\)</span> test statistic.
Here is what it should look like for the two counts in this dataset,</p>
<p><span class="math display">\[\chi^{2} = \frac{(O_{\mathrm{surv}} - E_{\mathrm{surv}})^{2}}{E_{\mathrm{surv}}} + \frac{(O_{\mathrm{died}} - E_{\mathrm{died}})^{2}}{E_{\mathrm{died}}}.\]</span>
What is the <span class="math inline">\(\chi^{2}\)</span> value?</p>
<p><span class="math inline">\(\chi^{2} =\)</span> _____________</p>
<p>There are two categories for survival (Yes and No).
How many degrees of freedom are there?</p>
<p><span class="math inline">\(df =\)</span> ________________</p>
<p>Now we can try to use jamovi to replicate the analysis above and find a p-value.
In the jamovi Analyses tab, select ‘Frequencies’ from the toolbar, then select ‘N Outcomes: <span class="math inline">\(\chi^{2}\)</span> Goodness of fit’ from the pull-down menu.
A new window will open up.</p>
<p>After selecting the option ‘N Outcomes: <span class="math inline">\(\chi^{2}\)</span> Goodness of fit’, a new window will appear called ‘Proportion Test (N Outcomes)’.
To run a <span class="math inline">\(\chi^{2}\)</span> goodness of fit test on bee survival, move the ‘survived’ variable into the ‘Variable’ box.
Leave the Counts box empty (Figure 31.1).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-133"></span>
<img src="img/jamovi_goodness_of_fit_interface.png" alt="Jamovi interface is shown with boxes of variables to be used for a proportion test, and the 'survived' variable has been placed in the Variable box." width="100%" />
<p class="caption">
Figure 31.1: Jamovi interface for running a Chi-square goodness of fit test on bumblebee survival in a dataset.
</p>
</div>
<p>The <span class="math inline">\(\chi^{2}\)</span> Goodness of Fit table will appear in the panel to the right.
From this table, we can see the <span class="math inline">\(\chi^{2}\)</span> test statistic, degrees of freedom (df), and the p-value (p).
Write these values below, and check to see if the <span class="math inline">\(\chi^{2}\)</span> and <span class="math inline">\(df\)</span> match the values you calculated above by hand.</p>
<p><span class="math inline">\(\chi^{2} =\)</span> _____________</p>
<p><span class="math inline">\(df =\)</span> ________________</p>
<p><span class="math inline">\(P =\)</span> ________________</p>
<p>Next, we will try another goodness of fit test, but this time to test whether or not bees were taken from all colonies with the same probability.</p>
</div>
<div id="colony-goodness-of-fit" class="section level2 hasAnchor" number="31.2">
<h2><span class="header-section-number">31.2</span> Colony goodness of fit<a href="Chapter_31.html#colony-goodness-of-fit" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Next, suppose that we want to know if bees were sampled from the colonies with the same expected frequencies.
What are the null and alternative hypotheses in this scenario?</p>
<ul>
<li><p><span class="math inline">\(H_{0}\)</span>: _________________</p></li>
<li><p><span class="math inline">\(H_{A}\)</span>: _________________</p></li>
</ul>
<p>How many colonies are there in this dataset?</p>
<p>Colonies: ________________</p>
<p>Run the <span class="math inline">\(\chi^{2}\)</span> goodness of fit test using the same procedure in jamovi that you used in the previous exercise.
What is the output from the Goodness of Fit table?</p>
<p><span class="math inline">\(\chi^{2} =\)</span> ____________</p>
<p><span class="math inline">\(df =\)</span> _____________</p>
<p><span class="math inline">\(P =\)</span> ____________</p>
<p>From this output, what can you conclude about how bees were taken from the colonies?</p>
<pre><code>
</code></pre>
<p>Note that the distrACTION module in jamovi includes a <span class="math inline">\(\chi^{2}\)</span> distribution (called ‘x2-Distribution’), which you can use to compute probabilities and quantiles in the same way we did for previous distributions in the book.
Next, we will move on to a <span class="math inline">\(\chi^{2}\)</span> test of association between colony and survival.</p>
</div>
<div id="chi-square-test-of-association-1" class="section level2 hasAnchor" number="31.3">
<h2><span class="header-section-number">31.3</span> Chi-Square test of association<a href="Chapter_31.html#chi-square-test-of-association-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Suppose we want to know if there is an association between bee colony and bee survival.
We can use a <span class="math inline">\(\chi^{2}\)</span> test of association to investigate this question.
What are the null and alternative hypotheses for this test of association?</p>
<ul>
<li><p><span class="math inline">\(H_{0}\)</span>: _________________</p></li>
<li><p><span class="math inline">\(H_{A}\)</span>: _________________</p></li>
</ul>
<p>To run the <span class="math inline">\(\chi^{2}\)</span> test of association, choose ‘Frequencies’ from the jamovi toolbar, but this time select ‘Independent Samples: <span class="math inline">\(\chi^{2}\)</span> test of association’ from the pull-down menu.
To test for an association between bee colony and survival, place ‘colony’ in the ‘Rows’ box and ‘survived’ in the ‘Columns’ box.
Leave the rest of the boxes blank (Figure 31.2).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-134"></span>
<img src="img/jamovi_test_of_association_interface.png" alt="Jamovi interface is shown with boxes of variables to be used for a Chi-square test of association with colony in a 'Rows' box and survived in a 'columns' box." width="100%" />
<p class="caption">
Figure 31.2: Jamovi interface for running a Chi-square test of association on bumblebee survival versus colony in a dataset.
</p>
</div>
<p>There is a pull-down called ‘Statistics’ below the Contingency Tables input.
Make sure that the <span class="math inline">\(\chi^{2}\)</span> checkbox is selected.
Output from the <span class="math inline">\(\chi^{2}\)</span> test of association will appear in the panel to the right.
Report the key statistics in the output table below.</p>
<p><span class="math inline">\(\chi^{2} =\)</span> ______________</p>
<p><span class="math inline">\(df =\)</span> _____________</p>
<p><span class="math inline">\(P =\)</span> _____________</p>
<p>From these statistics, should you reject or not reject the null hypothesis?</p>
<p><span class="math inline">\(H_{0}\)</span>: ______________</p>
<p>Note that scrolling down further in the left panel (Contingency Tables) reveals an option for plotting.
Have a look at this and create a barplot by checking ‘Bar Plot’ under <strong>Plots</strong>.
Note that there are various options for changing bar types (side by side or stacked), y-axis limits (counts versus percentages), and bar groupings (by rows or columns).</p>
<p>Now try running a <span class="math inline">\(\chi^{2}\)</span> test of association to see if there is an association between radiation and bee survival (hint, you just need to swap ‘colony’ for ‘radiation’ in the Rows box).
What can you conclude from this test?
Explain your conclusion as if you were reporting the results of the test to someone who was unfamiliar with statistical hypothesis testing.</p>
<pre><code>
</code></pre>
<p>Lastly, did the order in which you placed the two variables matter?
What if you switched Rows and Columns?
In other words, put ‘survived’ in the Rows box and ‘radiation’ in the Columns box.
Does this give you the same answer?</p>
<pre><code>
</code></pre>
<p>Next, we will look at correlations between variables.</p>
</div>
<div id="pearson-product-moment-correlation-test" class="section level2 hasAnchor" number="31.4">
<h2><span class="header-section-number">31.4</span> Pearson product moment correlation test<a href="Chapter_31.html#pearson-product-moment-correlation-test" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Suppose that we want to test if bumblebee mass at the start of the experiment (mass) is associated with carbon dioxide output (CO<sub>2</sub>).
Specifically, we want to know if more massive bees also output less carbon dioxide.
Before running any test, it is a good idea to plot the two variables using a scatterplot.
To do this, select the ‘Exploration’ button from the toolbar in jamovi, but instead of choosing ‘Descriptives’ as usual from the pull-down menu, select ‘Scatterplot’.
A new window will open up that allows you to build a scatterplot by selecting the variables that you place on the x-axis and the y-axis.
Put mass on the x-axis and CO<sub>2</sub> on the y-axis, as shown in Figure 31.3.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-135"></span>
<img src="img/jamovi_simple_scatterplot.png" alt="Jamovi interface is shown with a variable called 'mass' placed in a box called 'X-Axis', and a variable called 'CO_2' placed in a box called 'Y-axis'." width="100%" />
<p class="caption">
Figure 31.3: Jamovi interface for building a scatterplot with bumblebee mass on the x-axis and carbon dioxide output on the y-axis.
</p>
</div>
<p>Notice that the scatterplot appears in the panel on the right.
Each point in the scatterplot is a different bee (i.e., row).
Just looking at the scatterplot, does it appear as though bee mass and CO<sub>2</sub> output are correlated?
Why or why not?</p>
<pre><code>
</code></pre>
<p>Note that it is possible to separate points in the scatterplot by group.,
Try placing ‘survived’ in the box ‘Group’.</p>
<p>Now we can test whether or not bee mass and CO<sub>2</sub> output are negatively correlated.
What are the null and alternative hypotheses of this test?</p>
<ul>
<li><p><span class="math inline">\(H_{0}\)</span>: _________________</p></li>
<li><p><span class="math inline">\(H_{A}\)</span>: _________________</p></li>
</ul>
<p>Before we test whether or not the correlation coefficient (<span class="math inline">\(r\)</span>) is significant, we need to know which correlation coefficient to use.
Remember from <a href="Chapter_30.html#correlation-hypothesis-testing">Section 30.3</a> that a test of the Pearson product moment correlation assumes that the sample <span class="math inline">\(r\)</span> is normally distributed around the true correlation coefficient.
If both of our variables (mass and CO<sub>2</sub>) are normally distributed, then we can be confident that this assumption will not be violated.
But if one or both variables are not normally distributed, then we should consider using Spearman’s rank correlation coefficient instead.
To test if mass and CO<sub>2</sub> are normally distributed, navigate to the Descriptives panel in jamovi (where we usually find the summary statistics of variables).
Place mass and CO<sub>2</sub> in the ‘Variables’ box, then scroll down and notice that there is a checkbox under <strong>Normality</strong> for ‘Shapiro-Wilk’.
Check this box, then find the p-values for the Shapiro-Wilk test of normality in the panel to the right.
Write these p-values down below.</p>
<p>Mass: <span class="math inline">\(P =\)</span> _____________</p>
<p>CO<sub>2</sub>: <span class="math inline">\(P =\)</span> ___________</p>
<p>Based on these p-values, which type of correlation coefficient should we use to test <span class="math inline">\(H_{0}\)</span>, and why?</p>
<pre><code>
</code></pre>
<p>To run the correlation coefficient test, choose the button in the jamovi toolbar called ‘Regression’, then select the first option ‘Correlation Matrix’ from the pull-down menu.
The Correlation Matrix option will pull up a new window in jamovi (Figure 31.4).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-136"></span>
<img src="img/jamovi_correlation_interface.png" alt="Jamovi interface is shown with a variables called 'mass' and 'CO_2' placed in a box to the right and a checkbox for 'Pearson' selected" width="100%" />
<p class="caption">
Figure 31.4: Jamovi interface for testing correlation coefficients.
</p>
</div>
<p>Notice that the Pearson product moment correlation is selected in the checkbox of Figure 31.4 (‘Pearson’).
Immediately below this checkbox is a box called ‘Spearman’, which would report Spearman’s rank correlation coefficient test.
Below the <strong>Correlation Coefficients</strong> options, there are options for <strong>Hypothesis</strong>.
Remember that we are interested in the alternative hypothesis that mass and CO<sub>2</sub> are negatively correlated, so we should select the radio button ‘Correlated negatively’.</p>
<p>The output of the correlation test appears in the panel on the right in the form of a table called ‘Correlation Matrix’.
This table reports both the correlation coefficient (here called ‘Pearson’s r’) and the p-value.
Write these values below.</p>
<p><span class="math inline">\(r =\)</span> ______________</p>
<p><span class="math inline">\(P =\)</span> _____________</p>
<p>Based on this output, what should we conclude about the association between bumblebee mass and carbon dioxide output?</p>
<pre><code>
</code></pre>
<p>Next, we will test whether or not bee mass is associated with nectar consumption.</p>
</div>
<div id="spearmans-rank-correlation-test" class="section level2 hasAnchor" number="31.5">
<h2><span class="header-section-number">31.5</span> Spearman’s rank correlation test<a href="Chapter_31.html#spearmans-rank-correlation-test" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Next, we will test whether or not bee mass and nectar consumption are correlated.
What are the null and alternative hypotheses of this test?</p>
<ul>
<li><p><span class="math inline">\(H_{0}\)</span>: _________________</p></li>
<li><p><span class="math inline">\(H_{A}\)</span>: _________________</p></li>
</ul>
<p>Run a Shapiro-Wilk test of normality on each of the two variables, as was done in the previous exercise.
Based on the output of these tests, what kind of correlation coefficient should we use for testing the null hypothesis?</p>
<p>Correlation coefficient: _______________</p>
<p>Test whether or not bee mass and nectar consumption are correlated.
What is the correlation coefficient and p-value from this test?</p>
<p><span class="math inline">\(r =\)</span> ______________</p>
<p><span class="math inline">\(P =\)</span> _____________</p>
<p>Based on these results, should we reject or not reject the null hypothesis?</p>
<p><span class="math inline">\(H_{0}\)</span>: ____________</p>
<p>Suppose that we had used the Pearson product moment correlation coefficient instead of Spearman’s rank correlation coefficient.
Would we have made the same conclusion about the correlation (or lack thereof) between bee mass and nectar consumption?
Why or why not?</p>
<pre><code>
</code></pre>
</div>
<div id="untidy-goodness-of-fit" class="section level2 hasAnchor" number="31.6">
<h2><span class="header-section-number">31.6</span> Untidy goodness of fit<a href="Chapter_31.html#untidy-goodness-of-fit" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In Exercise 31.1, we ran a <span class="math inline">\(\chi^{2}\)</span> test using data in a tidy format, in which each row corresponded to a single observation, and categorical data were listed over <span class="math inline">\(N = 256\)</span> rows.
For the ‘survived’ variable, this meant 256 rows of ‘Yes’ or ‘No’.
But there is a shortcut in jamovi if we do not have a full tidy dataset.
If you know that the dataset included 139 ‘Yes’ counts and 117 ‘No’ counts, you could set up the data as a table of counts (Table 31.1).</p>
<table style="width:26%;">
<caption><strong>TABLE 31.1</strong> Counts of bees that did not survive (No) or did survive (Yes) in an experiment involving radiation.</caption>
<colgroup>
<col width="15%" />
<col width="11%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Survived</th>
<th align="center">Count</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">No</td>
<td align="center">117</td>
</tr>
<tr class="even">
<td align="center">Yes</td>
<td align="center">139</td>
</tr>
</tbody>
</table>
<p>Open a new data frame in jamovi, then recreate the small dataset in Table 31.1.
Column names should be ‘Survived’ and ‘Count’, as shown in Figure 31.5.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-138"></span>
<img src="img/jamovi_simple_counts.png" alt="Jamovi spreadsheet showing just 2 rows and 2 columns as 'Survived' and 'Count'." width="100%" />
<p class="caption">
Figure 31.5: Jamovi data frame with a simple organisation of count data.
</p>
</div>
<p>Next, navigate to the ‘Analyses’ tab and choose ‘N Outcomes’ to do a goodness of fit test.
Place ‘Survived’ in the Variable box, then place ‘Count’ in the Counts (optional) box.
Notice that you will get the same <span class="math inline">\(\chi^{2}\)</span>, <span class="math inline">\(df\)</span>, and p-values in the output table as you did in Exercise 31.1.</p>
<p>We could do the same for a <span class="math inline">\(\chi^{2}\)</span> test of association, although it would be a bit more complicated.
To test for an association between radiation and survival, as we did at the end of Exercise 31.3, we would need three columns and eight rows of data (Table 31.2).</p>
<table style="width:43%;">
<caption><strong>TABLE 31.2</strong> Counts of bees that did not survive (No) or did survive (Yes) for different levels of radiation.</caption>
<colgroup>
<col width="15%" />
<col width="16%" />
<col width="11%" />
</colgroup>
<thead>
<tr class="header">
<th align="center">Survived</th>
<th align="center">Radiation</th>
<th align="center">Count</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">No</td>
<td align="center">None</td>
<td align="center">12</td>
</tr>
<tr class="even">
<td align="center">Yes</td>
<td align="center">Low</td>
<td align="center">52</td>
</tr>
<tr class="odd">
<td align="center">No</td>
<td align="center">Medium</td>
<td align="center">29</td>
</tr>
<tr class="even">
<td align="center">Yes</td>
<td align="center">High</td>
<td align="center">35</td>
</tr>
<tr class="odd">
<td align="center">No</td>
<td align="center">None</td>
<td align="center">39</td>
</tr>
<tr class="even">
<td align="center">Yes</td>
<td align="center">Low</td>
<td align="center">25</td>
</tr>
<tr class="odd">
<td align="center">No</td>
<td align="center">Medium</td>
<td align="center">37</td>
</tr>
<tr class="even">
<td align="center">Yes</td>
<td align="center">High</td>
<td align="center">27</td>
</tr>
</tbody>
</table>
<p>If we put Table 31.2 into jamovi, we can run a <span class="math inline">\(\chi^{2}\)</span> test of association by navigating to the ‘Frequencies’ button in the jamovi toolbar and selecting ‘Independent Samples: <span class="math inline">\(\chi^{2}\)</span> test of association’ from the pull-down.
In the Contingency Tables input panel, we can put ‘Survived’ in the Rows box, ‘Radiation’ in the Columns box, then place ‘Count’ in the Counts (optional) box.
The panel on the right will give us the output of the <span class="math inline">\(\chi^{2}\)</span> test of association.</p>
</div>
</div>
<h3>References<a href="references.html#references" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body hanging-indent" line-spacing="2">
<div id="ref-Burrows2022" class="csl-entry">
Burrows, J. E., Copplestone, D., Raines, K. E., Beresford, N. A., & Tinsley, M. C. (2022). <span class="nocase">Ecologically relevant radiation exposure triggers elevated metabolic rate and nectar consumption in bumblebees</span>. <em>Functional Ecology</em>, <em>36</em>(8), 1822–1833. <a href="https://doi.org/10.1111/1365-2435.14067">https://doi.org/10.1111/1365-2435.14067</a>
</div>
<div id="ref-Jamovi2022" class="csl-entry">
The jamovi project. (2024). <em>Jamovi (version 2.5)</em>. <a href="https://www.jamovi.org">https://www.jamovi.org</a>
</div>
</div>
<div class="footnotes">
<hr />
<ol start="74">
<li id="fn74"><p><a href="https://bradduthie.github.io/stats/data/bumblebee.csv">https://bradduthie.github.io/stats/data/bumblebee.csv</a><a href="Chapter_31.html#fnref74" class="footnote-back">↩︎</a></p></li>
</ol>
</div>
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