【正文】 he degrees of freedom are n+m2 Case 2: ttest, pooled variances 222~ ????? mnppmntmsnsYXT2)1()1( 222??????mnsmsns yxpAlternate calculation formula: ttest, pooled variance 2~ ????? mnpmn tmnnmsYXT)()()11( 2222mnmnsmnmmnnsnmsnsmsppppp ???????Pooled vs. unpooled variance Rule of Thumb: Use pooled unless you have a reason not to. Pooled gives you more degrees of freedom. Pooled has extra assumption: variances are equal between the two groups. SAS automatically tests this assumption for you (“Equality of Variances” test). If p.05, this suggests unequal variances, and better to use unpooled ttest. Example: twosample ttest ? In 1980, some researchers reported that “men have more mathematical ability than women” as evidenced by the 1979 SAT’s, where a sample of 30 random male adolescents had a mean score 177。 pval=(1probt(.98, 58))*2。) Example 2: Difference in means ? Example: Rosental, R. and Jacobson, L. (1966) Teachers’ expectancies: Determinates of pupils’ . gains. Psychological Reports, 19, 115118. The Experiment (note: exact numbers have been altered) ? Grade 3 at Oak School were given an IQ test at the beginning of the academic year (n=90). ? Classroom teachers were given a list of names of students in their classes who had supposedly scored in the top 20 percent。 Oute Variable Are the observations independent or correlated? Alternatives if the normality assumption is violated (and small sample size): independent correlated Continuous (. pain scale, cognitive function) Ttest: pares means between two independent groups ANOVA: pares means between more than two independent groups Pearson’s correlation coefficient (linear correlation): shows linear correlation between two continuous variables Linear regression: multivariate regression technique used when the oute is continuous。 Oute Variable Are the observations independent or correlated? Alternatives if the normality assumption is violated (and small sample size): independent correlated Continuous (. pain scale, cognitive function) Ttest: pares means between two independent groups ANOVA: pares means between more than two independent groups Pearson’s correlation coefficient (linear correlation): shows linear correlation between two continuous variables Linear regression: multivariate regression technique used when the oute is continuous。 Olfson et al. Arch Gen 。 put pval。222222?????????nsnsnsnsnsyxtyyxxpypxpn2122112121 ??。 5. pvalue is sufficiently low for us to reject the null。=033.10.=ZWhen we ran this study 1000 times, we got 1 result as big or bigger than 10%. Pvalue from our simulation… We also got 3 results as small or smaller than –10%. Pvalue From our simulation, we estimate the pvalue to be: 4/1000 or .004 Here we reject the null. Alternative hypothesis: There is an association between antidepressant use and suicide in the target population. Hypothesis Testing Step 5: Reject or do not reject the null hypothesis. What would a lack of statistical significance mean? ? If this study had sampled only 50 cases and 50 controls, the sampling variability would have been much higher—as shown in this puter simulation… Standard error is about 10% 50 cases and 50 controls. Standard error is about % 263 cases and 1241 controls. With only 50 cases and 50 controls… Standard error is about 10% If we ran this study 1000 times, we would expect to get values of 10% or higher 170 times (or 17% of the time). Twotailed pvalue Twotailed pvalue = 17%x2=34% Practice problem… An August 2020 research article in Developmental and Behavioral Pediatrics reported the following about a sample of UK kids: when given a choice of a nonbranded chocolate cereal vs. CoCo Pops, 97% (36) of 37 girls and 71% (27) of 38 boys preferred the CoCo Pops. Is this evidence that girls are more likely to choose brandnamed products? Answer 1. Hypotheses: H0: p♂p♀= 0 Ha: p♂p♀≠ 0 [twosided] 2. Null distribution of difference of two proportions: 3. Observed difference in our experiment = .= .26 4. Calculate the pvalue of what you observed: 085.38)16(.84.37)16(.84.)38)75631(756337)75631(7563,0(~????????? ?Npp mfdata _null_。 gives rate of change over time Nonparametric statistics Wilcoxon signrank test: nonparametric alternative to the paired ttest Wilcoxon sumrank test (=MannWhitney U test): nonparametric alternative to the ttest KruskalWallis test: nonparametric alternative to ANOVA Spearman rank correlation coefficient: nonparametric alternative to Pearson’s correlation coefficient Nonparametric tests ? ttests require your oute variable to be normally distributed (or close enough), for small samples. ?