Inferential Statistics Psych 231: Research Methods in Psychology

Review from last time  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%  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%  Is the 4% difference a “real” difference (statistically significant) or is it just sampling error?

Testing Hypotheses 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 H 0 “statistically significant differences” Fail to reject H 0 “not statistically significant differences” Step 1: State your hypotheses Step 2: Set your decision criteria

Review from last time Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 Type I error Type II error  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%  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% XAXA XBXB 76%80%  Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Two sample distributions H 0 : there is no difference between Grp A and Grp B H 0 : μ A = μ B About populations H A : there is a difference between Grp A and Grp B

Testing Hypotheses 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 H 0 “statistically significant differences” Fail to reject H 0 “not statistically significant differences” Step 1: State your hypotheses Step 2: Set your decision criteria

Statistical significance “Statistically significant differences” When you “reject your null hypothesis” Essentially this means that the observed difference is larger than what you’d expect by chance With the probability of making a type I error at 5% (assuming an alpha level = 0.05) “Chance” is determined by estimating how much sampling error there is Factors affecting “chance” Population variability Sample size Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 Type I error Type II error 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

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 Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 Type I error Type II error

Testing Hypotheses 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 H 0 “statistically significant differences” Fail to reject H 0 “not statistically significant differences” Step 1: State your hypotheses Step 2: Set your decision criteria

“Generic” statistical test XBXB XAXA Why might the samples be different? (What is the source of the variability between groups)? 3 Sources: 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 Difference from chance = TR + ID + ER ID + ER = Computed test statistic

Statistical significance “Statistically significant differences” When you “reject your null hypothesis” Essentially this means that the observed difference is larger than what you’d expect by chance With the probability of making a type I error at 5% (assuming an alpha level = 0.05) “Chance” is determined by estimating how much sampling error there is Factors affecting “chance” Population variability Sample size

Sampling error: Pop var Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance” Small population variability Large population variability

Sampling error: sample size n = 1 Population mean x Sampling error (Pop mean - sample mean) Population Distribution

Sampling error: sample size n = 2 Population mean x Population Distribution x Sampling error (Pop mean - sample mean) Sample mean

Sampling error: sample size n = 10 Population mean Population Distribution Sampling error (Pop mean - sample mean) Sample mean x x x x x x x x x x  Generally, as the sample size increases, the sampling error decreases

Sampling error 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 XAXA XBXB XCXC XDXD Population Samples of size = n Distribution of sample means Avg. Sampling error “chance”

“Generic” statistical test Tests the question: Are there differences between groups due to a treatment? One population Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 Type I error Type II error H 0 is true (no treatment effect) XAXA XBXB 76%80% Two possibilities in the “real world” Two sample distributions

“Generic” statistical test XAXA XBXB XAXA XBXB H 0 is true (no treatment effect) H 0 is false (is a treatment effect) Two populations Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 Type I error Type II error 76%80% 76%80% People who get the treatment change, they form a new population (the “treatment population) People who get the treatment change, they form a new population (the “treatment population) Tests the question: Are there differences between groups due to a treatment? Two possibilities in the “real world”

“Generic” statistical test The generic test statistic distribution To reject the H 0, 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 α-level determines where these boundaries go Distribution of sample means Test statistic TR + ID + ER ID + ER

“Generic” statistical test Distribution of the test statistic Reject H 0 Fail to reject H 0 The generic test statistic distribution To reject the H 0, 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

“Generic” statistical test Distribution of the test statistic Reject H 0 Fail to reject H 0 The generic test statistic distribution To reject the H 0, 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 “One tailed test”: sometimes you know to expect a particular difference (e.g., “improve memory performance”)

“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

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

T-test Design 2 separate experimental conditions Degrees of freedom Based on the size of the sample and the kind of t-test Formula: T = X 1 - X 2 Diff by chance Based on sample error Observed difference Computation differs for between and within t-tests

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.”

Analysis of Variance Designs 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 XBXB XAXA XCXC

Analysis of Variance More than two groups 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 = XBXB XAXA XCXC

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

1 factor ANOVA Null hypothesis: H 0 : all the groups are equal X A = X B = X C Alternative hypotheses H A : not all the groups are equal X A ≠ X B ≠ X C X A ≠ X B = X C X A = X B ≠ X C X A = X C ≠ X B The ANOVA tests this one!! Do further tests to pick between these XBXB XAXA XCXC

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

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 < 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

Factorial ANOVAs 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 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