Statistics

1. Statistics Psych 231: Research Methods in Psychology

2. Quiz 10 (chapter 7) is due on Nov. 13th at midnight • Journal Summary 2 assignment • Due in class NEXT week (Wednesday, Nov. 18th) <- moved due date • Group projects • Plan to have your analyses done before Thanksgiving break, GAs will be available during lab times to help • Poster sessions are last lab sections of the semester (last week of classes), so start thinking about your posters. I will lecture about poster presentations on the Monday before Thanksgiving break. Reminders

3. Inferential statistics used to generalize back Population Sample • 2 General kinds of Statistics • Descriptive statistics • Used to describe, simplify, & organize data sets • Describing distributions of scores • Inferential statistics • Used to test claims about the population, based on data gathered from samples • Takes sampling error into account. Are the results above and beyond what you’d expect by random chance? Samples and Populations

4. Properties: Shape, Center, and Spread (variability) • Shape • Symmetric v. asymmetric (skew) • Unimodal v. multimodal • Center • Where most of the data in the distribution are • Mean, Median, Mode • Spread (variability) • How similar/dissimilar are the scores in the distribution? • Standard deviation (variance), Range Describing Distributions

5. Properties: Shape, Center, and Spread (variability) • Visual descriptions - A picture of the distribution is usually helpful • Numerical descriptions of distributions Describing Distributions

6. Divide by the total number in the population Add up all of the X’s Divide by the total number in the sample • The mean (mathematical average) is the most popular and most important measure of center. • The formula for the population mean is (a parameter): • The formula for the sample mean is (a statistic): mean Mean & Standard deviation

7. mean • The mean (mathematical average) is the most popular and most important measure of center. Others include median and mode. • The standard deviation is the most popular and important measure of variability. • The standard deviation measures how far off all of the individuals in the distribution are from a standard, where that standard is the mean of the distribution. • Essentially, the average of the deviations. Mean & Standard deviation

8. standard deviation = σ = • Working your way through the formula: • Step 1: Compute deviation scores • Step 2: Compute the SS • Step 3: Determine the variance • Take the average of the squared deviations • Divide the SS by the N • Step 4: Determine the standard deviation • Take the square root of the variance An Example: Computing Standard Deviation (population)

9. Main difference: • Step 1: Compute deviation scores • Step 2: Compute the SS • Step 3: Determine the variance • Take the average of the squared deviations • Divide the SS by the n-1 • Step 4: Determine the standard deviation • Take the square root of the variance • This is done because samples are biased to be less variable than the population. This “correction factor” will increase the sample’s SD (making it a better estimate of the population’s SD) An Example: Computing Standard Deviation (sample)

10. Inferential statistics used to generalize back Population Sample • 2 General kinds of Statistics • Descriptive statistics • Used to describe, simplify, & organize data sets • Describing distributions of scores • Inferential statistics • Used to test claims about the population, based on data gathered from samples • Takes sampling error into account. Are the results above and beyond what you’d expect by random chance? Statistics

11. Population Sample A Treatment X = 80% Sample B No Treatment X = 76% • Purpose: To make claims about populations based on data collected from samples • What’s the big deal? • 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? Inferential Statistics

12. Step 2: Set your decision criteria • Step 3: Collect your data from your sample(s) • Step 4: Compute your test statistics • Step 5: Make a decision about your null hypothesis • “Reject H0” • “Fail to reject H0” • Step 1: State your hypotheses Testing Hypotheses

13. This is the hypothesis that you are testing • Step 1: State your hypotheses • Null hypothesis (H0) • Alternative hypothesis(ses) • “There are no differences (effects)” • Generally, “not all groups are equal” • You aren’t out to prove the alternative hypothesis(although it feels like this is what you want to do) • If you reject the null hypothesis, then you’re left with support for the alternative(s) (NOT proof!) Testing Hypotheses

14. Step 1: State your hypotheses • In our memory example experiment • Null H0: mean of Group A = mean of Group B • Alternative HA: mean of Group A ≠ mean of Group B • (Or more precisely: Group A > Group B) • It seems like our theory is that the treatment should improve memory. • That’s the alternative hypothesis. That’s NOT the one the we’ll test with inferential statistics. • Instead, we test the H0 Testing Hypotheses

15. Step 2: Set your decision criteria • Your alpha level will be your guide for when to: • “reject the null hypothesis” • “fail to reject the null hypothesis” • Step 1: State your hypotheses • This could be correct conclusion or the incorrect conclusion • Two different ways to go wrong • Type I error: saying that there is a difference when there really isn’t one (probability of making this error is “alpha level”) • Type II error: saying that there is not a difference when there really is one Testing Hypotheses

16. Real world (‘truth’) H0 is correct H0 is wrong Type I error Reject H0 Experimenter’s conclusions Fail to Reject H0 Type II error Error types

17. Real world (‘truth’) Defendant is innocent Defendant is guilty Type I error Find guilty Jury’s decision Type II error Find not guilty Error types: Courtroom analogy

18. Type I error: concluding that there is an effect (a difference between groups) when there really isn’t. • Sometimes called “significance level” • We try to minimize this (keep it low) • Pick a low level of alpha • Psychology: 0.05 and 0.01 most common • For Step 5, we compare a “p-value” of our test to the alpha level to decide whether to “reject” or “fail to reject” to H0 • Type II error: concluding that there isn’t an effect, when there really is. • Related to the Statistical Power of a test • How likely are you able to detect a difference if it is there Error types

19. 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” • Make this decision by comparing your test’s “p-value” against the alpha level that you picked in Step 2. • Step 1: State your hypotheses • Step 2: Set your decision criteria Testing Hypotheses

20. Consider the results of our class experiment X • Main effect of cell phone ✓ • Main effect of site type ✓ • An Interaction between cell phone and site type 0.25 -0.36 Factorial designs Resource: Dr. Kahn's reporting stats page

21. “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 Statistical significance

22. Population mean Population Distribution Sampling error (Pop mean - sample mean) x n = 1 Sampling error

23. Population mean Population Distribution Sample mean Sampling error (Pop mean - sample mean) x x n = 2 Sampling error

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

25. Small population variability Large population variability • Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance” Sampling error

26. Population Distribution of sample means XB XC XD Avg. Sampling error “chance” XA • 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 Samples of size = n Sampling error

27. “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

28. 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

29. About populations Real world (‘truth’) H0 is correct H0 is wrong Type I error Reject H0 Experimenter’s conclusions Fail to Reject H0 Type II error 76% 80% XB XA • 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 • Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Two sample distributions From last time

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

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

32. XA XB • 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

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

34. Population Distribution of sample means XB XC XD Avg. Sampling error “chance” XA • 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 Samples of size = n Sampling error

35. Test statistic TR + ID + ER ID + ER • 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 Distribution of sample means α-level determines where these boundaries go “Generic” statistical test

36. 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

37. 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

38. 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

39. “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

40. 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

41. 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

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

43. 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

44. XA XC XB • 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 Analysis of Variance

45. Observed variance F-ratio = XA XC XB Variance from chance • 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 Analysis of Variance

46. XA XC XB • 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 1 factor ANOVA

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

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

49. 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

50. 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