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<!DOCTYPE html>
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<title>Chapter 21 What is hypothesis testing? | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
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<nav role="navigation">
<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>
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<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>
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<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>
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<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>
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<div id="Chapter_21" class="section level1 hasAnchor" number="21">
<h1><span class="header-section-number">Chapter 21</span> What is hypothesis testing?<a href="Chapter_21.html#Chapter_21" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>Statistical hypotheses are different from scientific hypotheses.
In science, a hypothesis should make some kind of testable statement about the relationship between two or more different concepts or observations <span class="citation">(<a href="#ref-Bouma2000" role="doc-biblioref">Bouma, 2000</a>)</span>.
For example, we might hypothesise that in a particular population of sparrows, juveniles that have higher body mass will also have higher survival rates.
In contrast, statistical hypotheses compare a sample outcome to the outcome predicted given a relevant statistical distribution <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
That is, we start with a hypothesis that our data are sampled from some distribution, then work out whether or not we should reject this hypothesis.
This concept is counter-intuitive, but it is absolutely fundamental for understanding the logic underlying most modern statistical techniques <span class="citation">(<a href="#ref-Greenland2016" role="doc-biblioref">Greenland et al., 2016</a>; <a href="#ref-Mayo1996" role="doc-biblioref">Mayo, 1996</a>; <a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>, including all subsequent chapters of this book, so we will focus on it here in-depth.
The most instructive way to explain the general idea is with the example of coin flips <span class="citation">(<a href="#ref-Mayo1996" role="doc-biblioref">Mayo, 1996</a>)</span>, as we looked at in <a href="Chapter_15.html#Chapter_15">Chapter 15</a>.</p>
<div id="how-ridiculous-is-our-hypothesis" class="section level2 hasAnchor" number="21.1">
<h2><span class="header-section-number">21.1</span> How ridiculous is our hypothesis?<a href="Chapter_21.html#how-ridiculous-is-our-hypothesis" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Imagine that a coin is flipped 100 times.
We are told that the coin is fair, meaning that there is an equal probability of it landing on heads or tails (i.e., the probability is 0.5 for both heads and tails in any given flip).
From <a href="#Chapter_15.html#binomial-distribution">Section 15.4.1</a>, recall that the number of times out of 100 that the coin flip comes up heads will be described by a binomial distribution.
The most probable outcome will be 50 heads and 50 tails, but frequencies that deviate from this perfect 50:50 ratio (e.g., 48 heads and 52 tails) are also expected to be fairly common (Figure 21.1).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-81"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-81-1.png" alt="A barplot is shown with 101 bars, which correspond to the number of times a coin flip lands on heads; the distribution takes a humped shape." width="672" />
<p class="caption">
Figure 21.1: Probability distribution for the number of times that a flipped coin lands on heads in 100 trials. Note that some areas of parameter space on the x-axis are cut off because the probabilities associated with this number of flips out of 100 being heads are so low.
</p>
</div>
<p>The distribution in Figure 21.1 is what we expect to happen if the coin we are flipping 100 times is actually fair.
In other words, it is the predicted distribution of outcomes <em>if our hypothesis that the coin is fair is true</em> (more on that later).
Now, suppose that we actually run the experiment; we flip the coin in question 100 times.
Perhaps we observe heads 30 times out of the 100 total flips.
From the distribution in Figure 21.1, this result seems <em>very</em> unlikely if the coin is actually fair.
If we do the maths, the probability of observing 30 heads or fewer (i.e., getting anywhere between 0 and 30 heads total) is only <span class="math inline">\(P = 0.0000392507\)</span>.
And the probability of getting this much of a deviation from 50 heads (i.e., either 20 less than or 20 more than 50) is <span class="math inline">\(P = 0.0000785014\)</span> (two times 0.0000392507, since the binomial distribution is symmetrical around 50).
This seems a bit ridiculous!
Do we <em>really</em> believe that the coin is fair if the probability of getting a result this extreme is so low?</p>
<p>Getting 30 head flips is maybe a bit extreme.
What if we flip the coin 100 times and get 45 heads?
In this case, if the coin is fair, then we would predict this number of heads or fewer with a probability of about <span class="math inline">\(P = 0.0967\)</span> (i.e., about 9.67% of the time, we would expect to get 45 or fewer heads).
And we would predict a deviation as extreme as 5 from the 50:50 ratio of heads to tails with a probability of about <span class="math inline">\(P = 0.193\)</span> (i.e., about 19.3% of the time, we would get 45 heads or fewer, or 55 heads or more).
This does not sound nearly so unrealistic.
If a fair coin will give us this much of a deviation from the expected 50 heads and 50 tails about 20% of the time, then perhaps our hypothesis is not so ridiculous, and we can conclude the coin is indeed fair.</p>
<p>How improbable does our result need to be to cause us to reject our hypothesis that the coin is fair?
There is no definitive answer to this question.
In the biological and environmental sciences, we traditionally use a probability of 0.05, but this threshold is completely arbitrary<a href="#fn40" class="footnote-ref" id="fnref40"><sup>40</sup></a>.
All it means is that we are willing to reject our hypothesis (i.e., declare the coin to be unfair) when it is actually true (i.e., the coin really <em>is</em> fair) about 5% of the time.
Note that we do need to decide on some finite threshold for rejecting our hypothesis because even extremely rare events, by definition, can sometimes happen.
In the case of 100 coin flips, there is always a small probability of getting <em>any</em> number of heads from a fair coin (although getting zero heads would be extraordinarily rare, <span class="math inline">\(P \approx 7.89 \times 10^{-31}\)</span>, i.e., a decimal followed by 30 zeros, then a 7).
We can therefore never be <em>certain</em> about rejecting or not rejecting the hypothesis that we have a fair coin.</p>
<p>This was a very concrete example intended to provide an intuitive way of thinking about hypothesis testing in statistics.
In the next section, we will look more generally at what hypothesis testing means in statistics and the terminology associated with it.
But everything that follows basically relies on the same general logic as the coin-flipping example here; <strong>if our hypothesis is true, then what is the probability of our result?</strong></p>
</div>
<div id="statistical-hypothesis-testing" class="section level2 hasAnchor" number="21.2">
<h2><span class="header-section-number">21.2</span> Statistical hypothesis testing<a href="Chapter_21.html#statistical-hypothesis-testing" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>A statistical test is used to decide if we should reject the hypothesis that some observed value or calculated statistic was sampled from a particular distribution <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
In the case of the coin example in the previous section, the observed value was the number of heads, and the distribution was the binomial distribution.
In other cases, we might, e.g., test the hypothesis that a value was sampled from a normal or t-distribution.
In all of these cases, the hypothesis that we are testing is the <strong>null hypothesis</strong>, which we abbreviate as <span class="math inline">\(H_{0}\)</span> (e.g., the coin is fair).
Typically, <span class="math inline">\(H_{0}\)</span> is associated with the lack of an interesting statistical pattern, such as when a coin is fair, when there is no difference between two groups of observations, or when two variables are not associated with each other.
This null hypothesis contrasts an <strong>alternative hypothesis</strong>, which we abbreviate as <span class="math inline">\(H_{A}\)</span> (e.g., the coin is not fair).
Alternative hypotheses are always defined by some relationship to <span class="math inline">\(H_{0}\)</span> <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
Typically, <span class="math inline">\(H_{A}\)</span> is associated with something interesting happening, such as a biased coin, a difference between groups of observations, or an association between two variables.
Table 21.1 below presents some null and alternative hypotheses that might be relevant in the biological or environmental sciences.</p>
<table>
<caption><strong>TABLE 21.1</strong> Hypothetical null and alternative hypotheses in the biological and environmental sciences.</caption>
<colgroup>
<col width="46%" />
<col width="53%" />
</colgroup>
<thead>
<tr class="header">
<th>Null Hypothesis <span class="math inline">\(H_{0}\)</span></th>
<th>Alternative Hypothesis <span class="math inline">\(H_{A}\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>There is no difference between juvenile and adult sparrow mortality</td>
<td>Mortality differs between juvenile and adult sparrows</td>
</tr>
<tr class="even">
<td>Amphibian body size does not change with increasing latitude</td>
<td>Amphibian body size increases with latitude</td>
</tr>
<tr class="odd">
<td>Soil nitrogen concentration does not differ between agricultural and non-agricultural fields</td>
<td>Soil nitrogen concentration is lower in non-agricultural fields</td>
</tr>
</tbody>
</table>
<p>Notice that alternative hypotheses can indicate direction (e.g., amphibian body size will increase with latitude, or nitrogen content will be lower in non-agricultural fields), or they can be non-directional (e.g., mortality will be different based on life-history stage).
When our alternative hypothesis indicates direction, we say that the hypothesis is <strong>one-sided</strong>.
This is because we are looking at one side of the null distribution.
In the case of our coin example, a one-sided <span class="math inline">\(H_{A}\)</span> might be that the probability of flipping heads is less than 0.5, meaning that we reject <span class="math inline">\(H_{0}\)</span> only given numbers on the left side of the distribution in Figure 21.1 (where the number of times a coin flip is heads is fewer than 50).
A different one-sided <span class="math inline">\(H_{A}\)</span> would be that the probability of flipping heads is greater than 0.5, in which case we would reject <span class="math inline">\(H_{0}\)</span> only given numbers on the right side of the distribution.
In contrast, when our alternative hypothesis does not indicate direction, we say that the hypothesis is <strong>two-sided</strong>.
This is because we are looking at both sides of the null distribution.
In the case of our coin example, we might not care in which direction the coin is biased (towards heads or tails), just that the probability of flipping heads does not equal 0.5.
In this case, we reject <span class="math inline">\(H_{0}\)</span> at both extremes of the distribution of Figure 21.1.</p>
</div>
<div id="p-values-false-positives-and-power" class="section level2 hasAnchor" number="21.3">
<h2><span class="header-section-number">21.3</span> P-values, false positives, and power<a href="Chapter_21.html#p-values-false-positives-and-power" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In our hypothetical coin-flipping example, we used <span class="math inline">\(P\)</span> to indicate the probability of getting a particular number of heads out of 100 total flips if our coin was fair.
This <span class="math inline">\(P\)</span> (sometimes denoted with a lower-case <span class="math inline">\(p\)</span>) is what we call a ‘p-value’.</p>
<blockquote>
<p><strong>A p-value is the probability of getting a result as or more extreme than the one observed assuming <span class="math inline">\(H_{0}\)</span> is true.<a href="#fn41" class="footnote-ref" id="fnref41"><sup>41</sup></a></strong></p>
</blockquote>
<p>This is separated and in bold because it is a very important concept in statistics, and it is one that is very, very easy to misinterpret<a href="#fn42" class="footnote-ref" id="fnref42"><sup>42</sup></a>.
A p-value is <em>not</em> the probability that the null hypothesis is true (we actually have no way of knowing this probability).
It is also not the probability that an alternative hypothesis is false (we have no way of knowing this probability either).
A p-value specifically <em>assumes that the null hypothesis is true</em>, then asks what the probability of an observed result would be <em>conditional upon this assumption</em>.
In the case of our coin flipping example, we cannot really know the probability that the coin is fair or unfair (depending on your philosophy of statistics, this might not even make conceptual sense).
But we can say that <strong>if</strong> the coin <strong>is</strong> fair, then an observation of <span class="math inline">\(\leq 45\)</span> would occur with a probability of <span class="math inline">\(P = 0.0967\)</span>.</p>
<p>Before actually calculating a p-value, we typically set a threshold level (<span class="math inline">\(\alpha\)</span>) below which we will conclude that our p-value is <strong>statistically significant</strong><a href="#fn43" class="footnote-ref" id="fnref43"><sup>43</sup></a>.
As mentioned in <a href="Chapter_21.html#how-ridiculous-is-our-hypothesis">Section 21.1</a>, we traditionally set <span class="math inline">\(\alpha= 0.05\)</span> in the biological and environmental sciences (although rarely <span class="math inline">\(\alpha = 0.01\)</span> is used).
This means that if <span class="math inline">\(P < 0.05\)</span>, then we reject <span class="math inline">\(H_{0}\)</span> and conclude that our observation is statistically significant.
It also means that even when <span class="math inline">\(H_{0}\)</span> really is true (e.g., the coin really is fair), we will mistakenly reject <span class="math inline">\(H_{0}\)</span> with a probability of 0.05 (i.e., 5% of the time).
This is called a <strong>Type I error</strong> (i.e., false positive), and it typically means that we will infer a pattern of some kind (e.g., difference between groups, or relationship between variables) where none really exists.
This is obviously an error that we want to avoid, which is why we set <span class="math inline">\(\alpha\)</span> to a low value.</p>
<p>In contrast, we can also fail to reject <span class="math inline">\(H_{0}\)</span> when <span class="math inline">\(H_{A}\)</span> is actually true.
That is, we might mistakenly conclude that there is no evidence to reject the null hypothesis when the null hypothesis really is false.
This is called a <strong>Type II error</strong>.
The probability that we commit a Type II error, i.e., that we fail to reject the null hypothesis when it is false, is given the symbol <span class="math inline">\(\beta\)</span>.
Since <span class="math inline">\(\beta\)</span> is the probability that we fail to reject <span class="math inline">\(H_{0}\)</span> when it is false, <span class="math inline">\(1 - \beta\)</span> is the probability that we <em>do</em> reject <span class="math inline">\(H_{0}\)</span> when it is false.
This <span class="math inline">\(1 - \beta\)</span> is the <strong>statistical power</strong> of a test.
Note that <span class="math inline">\(\alpha\)</span> and <span class="math inline">\(\beta\)</span> are not necessarily related to each other.
Our <span class="math inline">\(\alpha\)</span> is whatever we set it to be (e.g., <span class="math inline">\(\alpha = 0.05\)</span>).
But statistical power will depend on the size of the effect that we are measuring (e.g., how much bias there is in a coin if we are testing whether or not it is fair), and on the size of our sample.
Increasing our sample size will always increase our statistical power, i.e., our ability to reject the null hypothesis when it is really false.
Table 21.2 illustrates the relationship between whether or not <span class="math inline">\(H_{0}\)</span> is true, and whether or not we reject it.</p>
<table>
<caption><strong>TABLE 21.2</strong> Summary of Type I and Type II errors in relation to a null hypothesis (<span class="math inline">\(H_{0}\)</span>).</caption>
<thead>
<tr class="header">
<th></th>
<th>Do Not Reject <span class="math inline">\(H_{0}\)</span></th>
<th>Reject <span class="math inline">\(H_{0}\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><span class="math inline">\(H_{0}\)</span> is true</td>
<td>Correct decision</td>
<td>Type I error</td>
</tr>
<tr class="even">
<td><span class="math inline">\(H_{0}\)</span> is false</td>
<td>Type II error</td>
<td>Correct decision</td>
</tr>
</tbody>
</table>
<p>Note that we never <em>accept</em> a null hypothesis; we just fail to reject it.
Statistical tests are not really set up in a way that <span class="math inline">\(H_{0}\)</span> can be accepted<a href="#fn44" class="footnote-ref" id="fnref44"><sup>44</sup></a>.
The reason for this is subtle, but we can see the logic if we again consider the case of the fair coin.
If <span class="math inline">\(H_{0}\)</span> is true, then the probability of flipping heads is <span class="math inline">\(P(heads) = 0.5\)</span> (i.e., <span class="math inline">\(H_{0}:\:P(heads) = 0.5\)</span>).
But even if we fail to reject <span class="math inline">\(H_{0}\)</span>, this does not mean that we can conclude with any real confidence that our null hypothesis <span class="math inline">\(P(heads) = 0.5\)</span> is true.
What if we instead tested the null hypothesis that our coin was <em>very slightly</em> biased, such that <span class="math inline">\(H_{0}:\:P(heads) = 0.4999\)</span>?
If we failed to reject the null hypothesis that <span class="math inline">\(P(heads) = 0.5\)</span>, then we would probably also fail to reject a <span class="math inline">\(H_{0}\)</span> that <span class="math inline">\(P(heads) = 0.4999\)</span>.
There is no way to meaningfully distinguish between these two potential null hypotheses by just testing one of them.
We therefore cannot conclude that <span class="math inline">\(H_{0}\)</span> is correct; we can only find evidence to reject it.
In contrast, we can reasonably accept an alternative hypothesis <span class="math inline">\(H_{A}\)</span> when we reject <span class="math inline">\(H_{0}\)</span>.</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-Bouma2000" class="csl-entry">
Bouma, G. D. (2000). <em><span>The Research Process</span></em> (4th ed., p. 242). Oxford University Press, Oxford, UK.
</div>
<div id="ref-Greenland2016" class="csl-entry">
Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). <span class="nocase">Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations</span>. <em>European Journal of Epidemiology</em>, <em>31</em>(4), 337–350. <a href="https://doi.org/10.1007/s10654-016-0149-3">https://doi.org/10.1007/s10654-016-0149-3</a>
</div>
<div id="ref-Mayo1996" class="csl-entry">
Mayo, D. G. (1996). <em><span class="nocase">Error and the Growth of Experimental Knowledge</span></em> (p. 493). University of Chicago Press, Chicago, USA.
</div>
<div id="ref-Mayo2019" class="csl-entry">
Mayo, D. G. (2019). <span class="nocase">P-value thresholds: Forfeit at your peril</span>. <em>European Journal of Clinical Investigation</em>, <em>49</em>(10), 1–4. <a href="https://doi.org/10.1111/eci.13170">https://doi.org/10.1111/eci.13170</a>
</div>
<div id="ref-Mayo2021" class="csl-entry">
Mayo, D. G. (2021). Significance tests: Vitiated or vindicated by the replication crisis in psychology? <em>Review of Philosophy and Psychology</em>, <em>12</em>(1), 101–120. <a href="https://doi.org/10.1007/s13164-020-00501-w">https://doi.org/10.1007/s13164-020-00501-w</a>
</div>
<div id="ref-McShane2019" class="csl-entry">
McShane, B. B., Gal, D., Gelman, A., Robert, C., & Tackett, J. L. (2019). Abandon statistical significance. <em>American Statistician</em>, <em>73</em>, 235–245. <a href="https://doi.org/10.1080/00031305.2018.1527253">https://doi.org/10.1080/00031305.2018.1527253</a>
</div>
<div id="ref-Sokal1995" class="csl-entry">
Sokal, R. R., & Rohlf, F. J. (1995). <em><span>Biometry</span></em> (3rd ed., p. 887). W. H. Freeman & Company, New York, USA.
</div>
<div id="ref-Stanton-Geddes2014" class="csl-entry">
Stanton-Geddes, J., De Freitas, C. G., & De Sales Dambros, C. (2014). <span class="nocase">In defense of P values: Comment on the statistical methods actually used by ecologists</span>. <em>Ecology</em>, <em>95</em>(3), 637–642. <a href="https://doi.org/10.1890/13-1156.1">https://doi.org/10.1890/13-1156.1</a>
</div>
<div id="ref-Wasserstein2016" class="csl-entry">
Wasserstein, R. L., & Lazar, N. A. (2016). The ASA’s statement on p-values: Context, process, and purpose. <em>American Statistician</em>, <em>70</em>(2), 129–133. <a href="https://doi.org/10.1080/00031305.2016.1154108">https://doi.org/10.1080/00031305.2016.1154108</a>
</div>
</div>
<div class="footnotes">
<hr />
<ol start="40">
<li id="fn40"><p>I have heard many apocryphal stories about how a probability of 0.05 was decided upon, but I have no idea which, if any, of these stories are actually true.<a href="Chapter_21.html#fnref40" class="footnote-back">↩︎</a></p></li>
<li id="fn41"><p>Technically, it also assumes that all of the assumptions of the model underlying the hypothesis test are true, but we will worry about this later.<a href="Chapter_21.html#fnref41" class="footnote-back">↩︎</a></p></li>
<li id="fn42"><p>In fact, the p-value is so easy to misinterpret and so widely misused, that some scientists have called for them to be abandoned entirely <span class="citation">(<a href="#ref-Wasserstein2016" role="doc-biblioref">Wasserstein & Lazar, 2016</a>)</span>, but see <span class="citation">Stanton-Geddes et al. (<a href="#ref-Stanton-Geddes2014" role="doc-biblioref">2014</a>)</span> and <span class="citation">Mayo (<a href="#ref-Mayo2019" role="doc-biblioref">2019</a>)</span>.<a href="Chapter_21.html#fnref42" class="footnote-back">↩︎</a></p></li>
<li id="fn43"><p>Like p-values, setting thresholds below which we consider <span class="math inline">\(P\)</span> to be significant is at least somewhat controversial <span class="citation">(<a href="#ref-Mayo2021" role="doc-biblioref">Mayo, 2021</a>; <a href="#ref-McShane2019" role="doc-biblioref">McShane et al., 2019</a>)</span>. But the use of statistical significance thresholds is ubiquitous in the biological and environmental sciences, so we will use them throughout this book (it is important to understand them and interpret them).<a href="Chapter_21.html#fnref43" class="footnote-back">↩︎</a></p></li>
<li id="fn44"><p>Note that we might, for non-statistical reasons, conclude the absence of a particular phenomenon or relationship between observations. For example, following a statistical test, we might become convinced that a coin really is fair, or that there is no relationship between sparrow body mass and survival. But these are conclusions about scientific hypotheses, not statistical hypotheses.<a href="Chapter_21.html#fnref44" class="footnote-back">↩︎</a></p></li>
</ol>
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