Statistics for Terrified Biologists

Preface1 How to use this bookIntroductionThe text of the chaptersWhat should you do if you run into trouble?ElephantsThe numerical examples in the textBoxesSpare-time activitiesExecutive summariesWhy go to all that bother?The bibliography2 IntroductionWhat are statistics?NotationNotation for calculating the mean3 Summarizing variationIntroductionDifferent summaries of variationRangeTotal deviationMean deviationVarianceWhy n ?Why the squared deviations?The standard deviationThe next chapterSpare-time activities4 When are sums of squares NOT sums of squares?IntroductionCalculating machines offer a quicker method of calculating sums of squaresAdded squaresThe correction factorAvoid being confused by the term 'sum of squares'Summary of the calculator method of calculating down to standard deviationSpare-time activities5 The normal distributionIntroductionFrequency distributionsThe normal distributionWhat per cent is a standard deviation worth?Are the percentages always the same as these?Other similar scales in everyday lifeThe standard deviation as an estimate of the frequency of a number occurring in a sampleFrom per cent to probabilityExecutive summary - The standard deviationImportance of the standard deviation6 The relevance of the normal distribution to biological dataTo recapIs our observed distribution normal?Checking for normalityWhat can we do about a distribution that clearly is not normal?TransformationGrouping samplesDoing nothing!How many samples are needed?Factors affecting how many samples we should takeCalculating how many samples are needed7 Further calculations from the normal distributionIntroductionIs 'A' bigger than 'B'?The yardstick for decidingDerivation of the standard error of a difference between two meansStep - from variance of single data to variance of meansStep - from variance of single data to 'variance of differences'Step - the combination of Steps and; the standard error of difference between means (s.e.d.m.)Recap of the calculation of s.e.d.m. from the variance calculated from the individual valuesThe importance of the standard error of differences between meansSummary of this chapterSpare-time activitiesExecutive summary - Standard error of a difference between two means8 The t-testIntroductionThe principle of the t-testThe t-test in statistical termsWhy t?Tables of the t-distributionThe standard t-testThe procedureThe actual t-testt-Test for means associated with unequal variancesThe s.e.d.m. when variances are unequalA worked example of the t-test for means associated with unequal variancesThe paired t-testPair when possibleSpare-time activitiesExecutive Summary - The t-test9 One tail or two?IntroductionWhy is the analysis of variance F-test one-tailed?The two-tailed F-testHow many tails has the t-test?The final conclusion on number of tails10 Analysis of Variance - What is it? How does it work?IntroductionSums of squares in the Analysis of VarianceSome 'made-up' variation to analyze by AnovaThe sum of squares tableUsing Anova to sort out the variation in Table CPhasePhaseSqADS - an important acronymBack to the sum of squares tableHow well does the analysis reflect the input?End PhaseDegrees of freedom in AnovaThe completion of the End PhaseThe Variance RatioThe relationship between 't' and 'F'Constraints on the Analysis of VarianceAdequate size of experiment0Equality of variance between treatmentsTesting the homogeneity of varianceThe element of chance: randomizationComparison between treatment means in the Analysis ofVarianceThe Least Significant DifferenceA caveat about using the LSDExecutive summary - The principle of the analysis of varianceSqADS in the analysis of varianceSums of squares of deviations - how far we have come since Chapter!11 Experimental designs for analysis of varianceIntroductionFully randomizedData for analysis of a fully randomized experimentPrelimsPhasePhaseEnd PhaseRandomized blocksData for analysis of a randomized block experimentPrelimsPhasePhaseEnd PhaseIncomplete blocksLatin squareData for the analysis of a Latin squarePrelimsPhasePhaseEnd PhaseFurther comments on the Latin square designSplit plotSpare-time activitiesExecutive summary - Analysis of a randomized blockexperimentPhasePhaseEnd Phase12 Introduction to factorial experimentsWhat is a factorial experiment?InteractionIf there is no interactionWhat if there is interaction?How about a biological example?Measuring any interaction between factors is often the main/only purpose of an experimentHow does a factorial experiment change the form of the Analysis of Variance?Degrees of freedom for interactionsThe similarity between the 'residual' in Phase and the 'interaction' in PhaseSums of squares for interactions13 Factor factorial experimentsIntroductionAn example of a-factor experimentAnalysis of the-factor experimentPrelimsPhasePhaseEnd Phase (of Phase)PhaseEnd Phase (of Phase)Two important things to remember about factorials before tackling the next chapterAnalysis of factorial experiments with unequal replicationSpare-time activityExecutive summary - Analysis of a-factor randomized block experimentPhases and (see Executive Summary, page XXX)Phase (a new phase!) - The phase of treatment 'supertotals'End Phase14 Factorial experiments with more than two factorsIntroductionDifferent 'orders' of interactionExample of a-factor experimentPrelimsPhasePhasePhaseTo the End PhaseAddendum - Additional working of sum of squares calculationsSpare-time activity15 Factorial experiments with split plotsIntroductionDeriving the split plot design from the randomized block designDegrees of freedom in a split plot analysisMain plotsSub-plotsNumerical example of a split plot experiment and its analysisCalculating the sums of squaresEnd PhaseComparison of split plot and randomized block experimentUses of split plot designsSpare-time activity16 The t-test in the analysis of varianceIntroductionBrief recap of relevant earlier sections of this bookLeast Significant Difference testMultiple range testsOperating the multiple range testTesting differences between meansSuggested 'rules' for testing differences between meansPresentation of the results of tests of differences between meansThe results of the experiments analyzed by analysis of variance in Chapters1 - 15Spare-time activity17 Linear regression and correlationIntroductionCause and effectOther traps waiting for you to fall intoExtrapolating beyond the range of your dataIs a straight line appropriate?The distribution of variabilityRegressionIndependent and dependent variablesThe regression coefficient (b)Calculating the regression coefficient (b)The regression equationA worked example on some real dataThe data (Box7.2)Calculating the regression coefficient (b) - i.e. the slope of the regression lineCalculating the intercept (a)Drawing the regression lineTesting the significance of the slope (b) of the regressionHow well do the points fit the line? - the coefficient of determination (r2)CorrelationDerivation of the correlation coefficient (r)An example of correlationIs there a correlation line?Extensions of regression analysisNonlinear regressionMultiple linear regressionMultiple nonlinear regressionAnalysis of covarianceSpare-time activitiesExecutive summary - Linear regressionThe calculation of b using the 'Sum of cross products'The regression equationHow well do the points fit the line?18 Chi-square testsIntroductionWhen and where not to use chi2 The problem of low frequenciesYates' correction for continuityThe chi2 test for 'goodness of fit'The case of more than two classeschi2 with heterogeneityHeterogeneity chi2 analysis with 'covariance'Association (or contingency) chi2 2 contingency tableFisher's exact test for a tableLarger contingency tablesInterpretation of contingency tablesSpare-time activities19 Nonparametric methods (what are they?)DisclaimerIntroductionAdvantages and disadvantages of the two approachesWhere nonparametric methods scoreWhere parametric methods scoreSome ways data are organized for nonparametric testsThe sign testThe Kruskal - Wallis analysis of ranksKendall's rank correlation coefficientThe main nonparametric methods that are availableAppendix How many replicates?Appendix Statistical tablesAppendix Solutions to 'Spare-time activities'Appendix BibliographyIndex