Inferential Statistics Psych 231: Research Methods in Psychology

Testing Hypotheses Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data from your sample(s) Step 4: Compute your test statistics Descriptive statistics (means, standard deviations, etc.) Inferential statistics (t-tests, ANOVAs, etc.) Step 5: Make a decision about your null hypothesis Reject H0 “statistically significant differences” Fail to reject H0 “not statistically significant differences” Testing Hypotheses

“Generic” statistical test The generic test statistic distribution To reject the H0, you want a computed test statistics that is large reflecting a large Treatment Effect (TR) What’s large enough? The alpha level gives us the decision criterion TR + ID + ER ID + ER Distribution of the test statistic Test statistic Distribution of sample means α-level determines where these boundaries go “Generic” statistical test

Some inferential statistical tests 1 factor with two groups T-tests Between groups: 2-independent samples Within groups: Repeated measures samples (matched, related) 1 factor with more than two groups Analysis of Variance (ANOVA) (either between groups or repeated measures) Multi-factorial Factorial ANOVA Some inferential statistical tests

T-test Design Formula: Observed difference X1 - X2 T = 2 separate experimental conditions Degrees of freedom Based on the size of the sample and the kind of t-test Formula: Observed difference T = X1 - X2 Diff by chance Based on sample error Computation differs for between and within t-tests T-test

T-test Reporting your results The observed difference between conditions Kind of t-test Computed T-statistic Degrees of freedom for the test The “p-value” of the test “The mean of the treatment group was 12 points higher than the control group. An independent samples t-test yielded a significant difference, t(24) = 5.67, p < 0.05.” “The mean score of the post-test was 12 points higher than the pre-test. A repeated measures t-test demonstrated that this difference was significant significant, t(12) = 5.67, p < 0.05.” T-test

Analysis of Variance XB XA XC Designs Test statistic is an F-ratio More than two groups 1 Factor ANOVA, Factorial ANOVA Both Within and Between Groups Factors Test statistic is an F-ratio Degrees of freedom Several to keep track of The number of them depends on the design Analysis of Variance

Analysis of Variance More than two groups F-ratio = XB XA XC Now we can’t just compute a simple difference score since there are more than one difference So we use variance instead of simply the difference Variance is essentially an average difference Observed variance Variance from chance F-ratio = Analysis of Variance

1 factor ANOVA 1 Factor, with more than two levels XB XA XC Now we can’t just compute a simple difference score since there are more than one difference A - B, B - C, & A - C 1 factor ANOVA

1 factor ANOVA The ANOVA tests this one!! XA = XB = XC XA ≠ XB ≠ XC Null hypothesis: H0: all the groups are equal The ANOVA tests this one!! XA = XB = XC Do further tests to pick between these Alternative hypotheses HA: not all the groups are equal XA ≠ XB ≠ XC XA ≠ XB = XC XA = XB ≠ XC XA = XC ≠ XB 1 factor ANOVA

1 factor ANOVA Planned contrasts and post-hoc tests: - Further tests used to rule out the different Alternative hypotheses XA ≠ XB ≠ XC Test 1: A ≠ B XA = XB ≠ XC Test 2: A ≠ C XA ≠ XB = XC Test 3: B = C XA = XC ≠ XB 1 factor ANOVA

1 factor ANOVA Reporting your results The observed differences Kind of test Computed F-ratio Degrees of freedom for the test The “p-value” of the test Any post-hoc or planned comparison results “The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < 0.05. Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(1) = 5.67, p < 0.05 & t(1) = 6.02, p <0.05). Groups B and C did not differ significantly from one another” 1 factor ANOVA

We covered much of this in our experimental design lecture More than one factor Factors may be within or between Overall design may be entirely within, entirely between, or mixed Many F-ratios may be computed An F-ratio is computed to test the main effect of each factor An F-ratio is computed to test each of the potential interactions between the factors Factorial ANOVAs

Factorial ANOVAs Reporting your results The observed differences Because there may be a lot of these, may present them in a table instead of directly in the text Kind of design e.g. “2 x 2 completely between factorial design” Computed F-ratios May see separate paragraphs for each factor, and for interactions Degrees of freedom for the test Each F-ratio will have its own set of df’s The “p-value” of the test May want to just say “all tests were tested with an alpha level of 0.05” Any post-hoc or planned comparison results Typically only the theoretically interesting comparisons are presented Factorial ANOVAs