Kin 304 Inferential Statistics

1. Kin 304Inferential Statistics Probability Level for Acceptance Type I and II Errors One and Two-Tailed tests Critical value of the test statistic “Statistics means never having to say you're certain”

2. Inferential Statistics • As the name suggests Inferential Statistics allow us to make inferences about the population, based upon the sample, with a specified degree of confidence Inferential Statistics

3. The Scientific Method • Select a sample representative of the population. The method of sample selection is crucial to this process along with the sample size being large enough to allow appropriate probability testing. • Calculate the appropriate test statistic. The test statistic used is determined by the hypothesis being tested and the research design as a whole. • Test the Null hypothesis. Compare the calculated test statistic to its critical value at the predetermined level of acceptance. Inferential Statistics

4. Setting a Probability Level for Acceptance • Prior to analysis the researcher must decide upon their level of acceptance. • Tests of significance are conducted at pre-selected probability levels, symbolized by p or α. • The vast majority of the time the probability level of 0.05, is used. • A p of .05 means that if you reject the null hypothesis, then you expect to find a result of this magnitude by chance only 5 in 100 times. Or conversely, if you carried out the experiment 100 times you would expect to find a result of this magnitude 95 times. You therefore have 95% confidence in your result. A more stringent test would be one where the p = 0.01, which translates to 99% confidence in the result. Inferential Statistics

5. No Rubber Yard Sticks • Either the researcher should pre-select one level of acceptance and stick to it, or do away with a set level of acceptance all together and simply report the exact probability of each test statistic. • If for instance, you had calculated a t statistic and it had an associated probability of p = 0.032, you could either say the probability is lower than the pre-set acceptance level of 0.05 therefore a significant difference at the 95% level of confidence or simply talk about 0.032 as a percentage confidence (96.8%) Inferential Statistics

6. Significance of Statistical Tests • The test statistic is calculated • The critical value of the test statistic is determined • based upon sample size and probability acceptance level (found in a table at the back of a stats book or part of the EXCEL stats report, or SPSS output) • The calculated test statistics must be greater than the critical value of the test statistic to accept a significant difference or relationship Inferential Statistics

8. Kin 304Tests of Differences between Means: t-tests SEM Visual test of differences Independent t-test Paired t-test

9. Comparison • Is there a difference between two or more groups? • Test of difference between means • t-test • (only two means, small samples) • ANOVA - Analysis of Variance • Multiple means • ANCOVA • covariates t Tests

10. Standard Error of the Mean Describes how confident you are that the mean of the sample is the mean of the population t Tests

11. Visual Test of Significant Difference between Means 1 Standard Error of the Mean 1 Standard Error of the Mean Overlapping standard error bars therefore no significant difference between means of A and B A B Mean No overlap of standard error bars therefore a significant difference between means of A and B at about 95% confidence

12. Independent t-test • Two independent groups compared using an independent T-Test (assuming equal variances) • e.g. Height difference between men and women • The t statistic is calculated using the difference between the means in relation to the variance in the two samples • A critical value of the t statistic is based upon sample size and probability acceptance level (found in a table at the back of a stats book or part of the EXCEL t-test report, or SPSS output) • the calculated t based upon your data must be greater than the critical value of t to accept a significant difference between means at the chosen level of probability t Tests

13. t statistic quantifiesthe degree of overlap of the distributions t Tests

14. standard error of the difference between means • The variance of the difference between means is the sum of the two squared standard deviations. • The standard error (S.E.) is then estimated by adding the squares of the standard deviations, dividing by the sample size and taking the square root. t Tests

15. t statistic • The t statistic is then calculated as the ratio of the difference between sample means to the standard error of the difference, with the degrees of freedom being equal to n - 2. t Tests

16. Critical values of t • Hypothesis: • There is a difference between means • Degrees of Freedom = 2n – 2 • tcalc > tcrit = significant difference t Tests

17. Paired Comparison • Paired t Test • sometimes called t-test for correlated data • “Before and After” Experiments • Bilateral Symmetry • Matched-pairs data t Tests

18. Paired t-test • Hypothesis: • Is the mean of the differences between paired observations significantly different than zero • the calculated t statistic is evaluated in the same way as the independent test t Tests

19. 9 Subjects All Lose Weight Mean of differences = +1.13

22. Kin 304Tests of Differences between Means:ANOVA – Analysis of Variance One-way ANOVA

23. ANOVA – Analysis of Variance • Used for analysis of multiple group means • Similar to independent t-test, in that the difference between means is evaluated based upon the variance about the means. • However multiple t-tests result in an increased chance of type 1 error. • F (ratio) statistic is calculated and is evaluated in comparison to the critical value of F (ratio) statistic Tests of Difference – ANOVA

24. One-way ANOVA • One grouping factor • HO: The population means are equal • HA: At least one group mean is different • Two or more levels of grouping factor • Exposure = low, medium or high • Age Groups = 7-8, 9-10, 11-12, 13-14 Tests of Difference – ANOVA

25. F (ratio) Statistic • The F ratio compares two sources of variability in the scores. • The variability among the sample means, called Between Group Variance, is compared with the variability among individual scores within each of the samples, called Within Group Variance. Tests of Difference – ANOVA

26. Formula for sources of variation Tests of Difference – ANOVA

27. Anova Summary Table Tests of Difference – ANOVA

28. Assumptions for ANOVA • The populations from which the samples were obtained are approximately normally distributed. • The samples are independent. • The population value for the standard deviation between individuals is the same in each group. • If standard deviations are unequal transformation of values may be needed. Tests of Difference – ANOVA

29. CFS Kids 17 – 19 years (Boys) • ANOVA • Dependent - VO2max • Grouping Factor - Age (17, 18, 19) • No Significant difference between means for VO2max (p>0.05)

30. CFS Kids 17 – 19 years (Girls) • ANOVA • Dependent - VO2max • Grouping Factor - Age (17, 18, 19) • Significant difference between means for VO2max (p<0.05)

31. Post Hoc tests • Post hoc simply means that the test is a follow-up test done after the original ANOVA is found to be significant. • One can do a series of comparisons, one for each two-way comparison of interest. • E.g. Scheffe or Tukey’s tests • The Scheffe test is very conservative Tests of Difference – ANOVA

32. Scheffe’s – Post Hoc Test Boys • Boys – no significant differences, would not run post hoc tests • Girls – VO2max for age19 is significantly different than at age17 Girls

33. ANOVA – Factorial designMultiple factors • Test of differences between means with two or more grouping factors, such that each factor is adjusted for the effect of the other • Can evaluate significance of factor effects and interactions between them • 2 – way ANOVA: Two factors considered simultaneously Tests of Difference – ANOVA

34. Example: 2 way ANOVA • Dependent - VO2max • Grouping Factors • AGE (17, 18, 19) • SEX (1, 2) • Significant difference in VO2max (p<0.05) by SEX=Main effect • Significant difference in VO2max (p<0.05) by AGE=Main effect • No Significant Interaction (p<0.05) AGE * SEX

35. Analysis of Covariance (ANCOVA) • Taking into account a relationship of the dependent with another continuous variable (covariate) in testing the difference between means of one or more factor • Tests significance of difference between regression lines Tests of Difference – ANOVA

36. Scatterplot showing correlations between skinfold-adjusted Forearm girth and maximum grip strength for men and women

37. Use of T tests for difference between means? • Men are significantly (p<0.05) bigger than women in skinfold-adjusted forearm girth and grip strength

38. ANCOVADependent – Maximum Grip Strength (GRIPR)Grouping Factor – Sex Covariate – Skinfold-adjusted Forearm Girth (SAFAGR) • SAFAGR is a significant Covariate • No significant difference between sexes in Grip Strength when adjusted for Covariate (representing muscle size) • Therefore one regression line (not two, for each sex) fit the relationship

39. 3-way ANOVA • For 3-way ANOVA, there will be: - three 2-way interactions (AxB, AxC) (BxC) - one 3-way interaction (AxBxC) • If for each interaction (p > 0.05) then use main effects results • Typically ANOVA is used only for 3 or less grouping factors Tests of Difference – ANOVA

40. Repeated Measures ANOVA • Repeated measures design – the same variable is measured several times over a period of time for each subject • Pre- and post-test scores are the simplest design – use paired t-test • Advantage - using fewer experimental units (subjects) and providing a control for differences (effect of variability due to differences between subjects can be eliminated) Tests of Difference – ANOVA