Psych 231: Research Methods in Psychology Statistics Psych 231: Research Methods in Psychology

Announcements Tomorrow’s office hours Cancelled Readings: Required: chapter 7 Recommended: chapter 14 (covers some specific hypothesis tests and reviews some of the SPSS how-to’s) Announcements

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” When you “reject your null hypothesis” Essentially this means that the observed difference is above what you’d expect by chance Testing Hypotheses

From last time XA XB About populations Example Experiment: Group A - gets treatment to improve memory Group B - gets no treatment (control) After treatment period test both groups for memory Results: Group A’s average memory score is 80% Group B’s is 76% H0: μA = μB H0: there is no difference between Grp A and Grp B Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Two sample distributions XA XB 76% 80% From last time

Testing for statistical significance The generic test statistic - is a ratio of sources of variability Observed difference 80% - 76% Computed test statistic = Difference from chance ?? “Statistically significant differences” When you “reject your null hypothesis” Essentially this means that the observed difference is above what you’d expect by chance “Chance” is determined by estimating how much sampling error there is Factors affecting “chance” Sample size Population variability Testing for statistical significance

Sample size on Sampling error Population mean x Sampling error (Pop mean - sample mean) Population Distribution n = 2 Population mean x Population Distribution Sampling error (Pop mean - sample mean) Sample mean n = 10 Population mean Population Distribution Sampling error (Pop mean - sample mean) Sample mean x Generally, as the sample size (n) increases, the “chance” (sampling error) decreases Sample size on Sampling error

Pop. Var. on Sampling error Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance” (sampling error) Large population variability Small population variability Pop. Var. on Sampling error

Sampling error Population Distribution of sample means These two factors combine to impact the distribution of sample means. The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population Population Distribution of sample means XC Samples of size = n XA XD Avg. Sampling error XB “chance” Sampling error

“Generic” statistical test Tests the question: Are there differences between groups due to a treatment? H0 is true (no treatment effect) Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Two possibilities in the “real world” One population Two sample distributions XA XB 76% 80% “Generic” statistical test

“Generic” statistical test Tests the question: Are there differences between groups due to a treatment? Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Two possibilities in the “real world” H0 is true (no treatment effect) H0 is false (is a treatment effect) Two populations: No treatment Two populations: Treatment XA XB XB XA 76% 80% 76% 80% People who get the treatment change, they form a new population (the “treatment population) “Generic” statistical test

“Generic” statistical test XB XA ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment Why might the samples be different? (What is the source of the variability between groups)? “Generic” statistical test

“Generic” statistical test XB XA ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment The generic test statistic - is a ratio of sources of variability Observed difference TR + ID + ER ID + ER Computed test statistic = = Difference from chance “Generic” statistical test

“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

“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 Distribution of the test statistic Reject H0 Fail to reject H0 “Generic” statistical test

“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 Distribution of the test statistic Reject H0 “One tailed test”: sometimes you know to expect a particular difference (e.g., “improve memory performance”) Fail to reject H0 “Generic” statistical test

“Generic” statistical test Things that affect the computed test statistic Size of the treatment effect The bigger the effect, the bigger the computed test statistic Difference expected by chance (sample error) Sample size Variability in the population “Generic” statistical test

Significance “A statistically significant difference” means: the researcher is concluding that there is a difference above and beyond chance with the probability of making a type I error at 5% (assuming an alpha level = 0.05) Note “statistical significance” is not the same thing as theoretical significance. Only means that there is a statistical difference Doesn’t mean that it is an important difference Significance

Non-Significance Failing to reject the null hypothesis Generally, not interested in “accepting the null hypothesis” (remember we can’t prove things only disprove them) Usually check to see if you made a Type II error (failed to detect a difference that is really there) Check the statistical power of your test Sample size is too small Effects that you’re looking for are really small Check your controls, maybe too much variability Non-Significance

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