Statistics Lecture 8: Independent & Paired T-tests

Lecture 8: Independent & Paired T-tests, Chi-Square Test 2 Types of T-Tests 1.​ Independent t-test a.​ Purpose: to test the difference between means when from 2 separate normally or “approximately” normally distributed populations 2.​ Paired t-test a.​ Purpose: comparing means between 2 matched or repeated measures from a normally distributed population Independent T-Test -​ W.S. Gosset and Guiness -​ Used to test the difference between means from 2 normally (or almost normally) distributed populations -​ Samples should be independent samples -​ Study Design: -​ Between-group -​ Independent design -​ Assumptions of T-test -​ Random sampling -​ The independent variable is nominal or categorical and contains 2 levels (you have 2 mutually exclusive groups of subjects) -​ Assumption of Independence: i.e. subjects contribute data/observations to 1 group ONLY -​ Sampling is from 2 normally distributed populations -​ There is a homogeneity of variance (the variances of the 2 populations are roughly equal) Steps to Set up Hypothesis to Compare 2 Population Means 1.​ Null and alternative hypothesis 2.​ Determine level of significance (a) 3.​ Run test statistic 4.​ Check the critical value for the significance level a and df = n1 + n2 - 2 a.​ (reject null if t statistic is equal or greater than the critical value) b.​ First must determine the degrees of freedom associated with the t statistic (look it up on a table) i.​ Based on the number of means calculated ii.​ Sample size for both groups minus 2 iii.​ degree of freedom = (n1 + n2) - 2 5.​ Are the results significant Mann-Whitney U Test -​ Non-parametric equivalent of the independent t-test -​ Uses ranks to determine if two independent samples were selected from populations that have the same distributions -​ Based on distribution of the ranks of data -​ Compares distribution (not means) of 2 independent samples -​ Generates a “U” statistic -​ Assumptions: -​ The independent variable is dichotomous -​ The 2 groups are independent -​ Dependent variable is at least ordinal -​ Total sample size of 8 or more (no less than 8) Paired T-Tests -​ Dependent means t-test, matched pairs, paired sample t-test -​ The measurement of 2 variables at 2 different points are compared -​ OR when data comes from same set of subjects at 2 different times -​ Study Design: -​ Repeated measures -​ Pre-test, post-test -​ Within subjects -​ Assumptions: -​ Two groups are dependent or paired -​ The distribution of differences is normal or approximately normally distributed -​ The sample/samples are random Steps of Paired T-Test 1.​ One-sided and two-sided test → ​ ​ ud = mean difference 2.​ Level of significance (a) 3.​ Run test statistic 4.​ Check critical value for significance level 5.​ Conclusion (significant of non-significant) Non parametric equivalent of a paired t-test: Wilcoxon matched pairs test Chi-Square Test -​ Used to test the independence of two variables or to test is there is a significant association between X and Y -​ Assesses the relationship between categorical variables: -​ Non-parametric test -​ Based on frequencies -​ Nominal or ordinal data -​ Two main types: -​ Chi square test of independence (to see if two variables are related) -​ Chi square goodness of fit (check sample data if it fits a specific distribution) -​ Limitations: -​ Highly sensitive to sample size -​ Sensitive to small frequencies -​ Little info on strength of relationship -​ Fisher’s Exact test: small sample when observed in one cell <5 -​ Yates Correction: more than 2 x 2 table (testing for independence and contingency) -​ Goodness of Fit Test: determine if sample matches a population -​ Characteristics: 1.​ The chi-square distribution is a family of curves based on the degrees of freedom 2.​ The chi-square distributions are positively skewed 3.​ All chi-square values are greater than or equal to zero 4.​ The total area under each distribution is equal to 1 -​ Assumptions: -​ Independent observations -​ Each subject should not be influenced by another subject in the study -​ Each subject should only contribute one observation in a study -​ Expected frequencies must be greater than 5 for at least 80% of the expected frequencies -​ If frequency less than 5: use fisher's exact test The Chi-Square Test for Goodness of Fit -​ Used to test the claim that an observed frequency distribution fit some given expected frequency distribution -​ Observed vs. expected -​ The data called observed frequencies simply count how many individuals from the sample are in each category Steps for Chi-Square 1.​ State hypothesis and identify the claim 2.​ Find the critical value from table G critical values (it is always right-tailed) 3.​ Compute the test value. Find the sum of the values 4.​ Make the decision 5.​ Summarize the results Fisher's Exact Test and Chi-Square Goodness of Fit Test -​ If the proportions in the cells are low then you need to use a variant of chi-square → the fisher's exact test -​ Fisher's exact test is more accurate than chi-square when numbers are small -​ Standard is for this to be if count is less than 5 in one of the cells