1. Statistics Psych 231: Research Methods in Psychology

2. Statistics 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

3. Statistics Why do we use them? 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

4. Inferential Statistics 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?  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?

5. Testing Hypotheses 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 H 0 ” “Fail to reject H 0 ” Step 1: State your hypotheses

6. Testing Hypotheses Step 1: State your hypotheses “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!) This is the hypothesis that you are testing Null hypothesis (H 0 ) Alternative hypothesis(ses)

7. Testing Hypotheses Step 1: State your hypotheses  In our memory example experiment  Null H 0 : mean of Group A = mean of Group B  Alternative H A : 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 H 0  In our memory example experiment  Null H 0 : mean of Group A = mean of Group B  Alternative H A : 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 H 0

8. Testing Hypotheses 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

9. Error types 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

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

11. Error types 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 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

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

13. “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

14. “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”

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

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

17. Statistical significance “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

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

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

20. Sampling error 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

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

22. 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”

23. “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

24. “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

25. “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”)

26. 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

27. 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

28. Summary 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

29. 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

30. 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

31. 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.”

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. Non-Experimental designs Sometimes you just can’t perform a fully controlled experiment Because of the issue of interest Limited resources (not enough subjects, observations are too costly, etc). Surveys Correlational Quasi-Experiments Developmental designs Small-N designs This does NOT imply that they are bad designs Just remember the advantages and disadvantages of each

41. Quasi-experiments What are they? Almost “true” experiments, but with an inherent confounding variable General types An event occurs that the experimenter doesn’t manipulate Something not under the experimenter’s control (e.g., flashbulb memories for traumatic events) Interested in subject variables high vs. low IQ, males vs. females Time is used as a variable

42. Quasi-experiments Program evaluation – Research on programs that is implemented to achieve some positive effect on a group of individuals. – e.g., does abstinence from sex program work in schools – Steps in program evaluation – Needs assessment - is there a problem? – Program theory assessment - does program address the needs? – Process evaluation - does it reach the target population? Is it being run correctly? – Outcome evaluation - are the intended outcomes being realized? – Efficiency assessment- was it “worth” it? The the benefits worth the costs?

43. Quasi-experiments Nonequivalent control group designs with pretest and posttest (most common) (think back to the second control lecture) participants Experimental group Control group Measure Non-Random Assignment Independent Variable Dependent Variable Measure Dependent Variable – But remember that the results may be compromised because of the nonequivalent control group (review threats to internal validity)

44. Quasi-experiments Advantages Allows applied research when experiments not possible Threats to internal validity can be assessed (sometimes) Disadvantages Threats to internal validity may exist Designs are more complex than traditional experiments Statistical analysis can be difficult Most statistical analyses assume randomness

45. Non-Experimental designs Sometimes you just can’t perform a fully controlled experiment Because of the issue of interest Limited resources (not enough subjects, observations are too costly, etc). Surveys Correlational Quasi-Experiments Developmental designs Small-N designs This does NOT imply that they are bad designs Just remember the advantages and disadvantages of each

46. Developmental designs Used to study changes in behavior that occur as a function of age changes Age typically serves as a quasi-independent variable Three major types Cross-sectional Longitudinal Cohort-sequential

47. Developmental designs Cross-sectional design Groups are pre-defined on the basis of a pre- existing variable Study groups of individuals of different ages at the same time Use age to assign participants to group Age is subject variable treated as a between-subjects variable Age 4 Age 7 Age 11

48. Cross-sectional design Developmental designs Advantages: Can gather data about different groups (i.e., ages) at the same time Participants are not required to commit for an extended period of time

49. Cross-sectional design Developmental designs Disavantages: Individuals are not followed over time Cohort (or generation) effect: individuals of different ages may be inherently different due to factors in the environment Are 5 year old different from 15 year olds just because of age, or can factors present in their environment contribute to the differences? Imagine a 15yr old saying “back when I was 5 I didn’t have a Wii, my own cell phone, or a netbook” Does not reveal development of any particular individuals Cannot infer causality due to lack of control

50. Longitudinal design Developmental designs Follow the same individual or group over time Age is treated as a within-subjects variable Rather than comparing groups, the same individuals are compared to themselves at different times Changes in dependent variable likely to reflect changes due to aging process Changes in performance are compared on an individual basis and overall Age 11 time Age 20Age 15

51. Longitudinal Designs Example Wisconsin Longitudinal Study (WLS) Wisconsin Longitudinal Study Began in 1957 and is still on-going (50 years) 10,317 men and women who graduated from Wisconsin high schools in 1957 Originally studied plans for college after graduation Now it can be used as a test of aging and maturation

52. Longitudinal design Developmental designs Advantages: Can see developmental changes clearly Can measure differences within individuals Avoid some cohort effects (participants are all from same generation, so changes are more likely to be due to aging)

53. Longitudinal design Developmental designs Disadvantages Can be very time-consuming Can have cross-generational effects: Conclusions based on members of one generation may not apply to other generations Numerous threats to internal validity: Attrition/mortality History Practice effects Improved performance over multiple tests may be due to practice taking the test Cannot determine causality

54. Developmental designs Measure groups of participants as they age Example: measure a group of 5 year olds, then the same group 10 years later, as well as another group of 5 year olds Age is both between and within subjects variable Combines elements of cross-sectional and longitudinal designs Addresses some of the concerns raised by other designs For example, allows to evaluate the contribution of cohort effects Cohort-sequential design

55. Developmental designs Cohort-sequential design Time of measurement 197519851995 Cohort A Cohort B Cohort C Cross-sectional component 1970s 1980s 1990s Age 5 Age 15Age 25 Age 5 Age 15 Age 5 Longitudinal component

56. Developmental designs Advantages: Get more information Can track developmental changes to individuals Can compare different ages at a single time Can measure generation effect Less time-consuming than longitudinal (maybe) Disadvantages: Still time-consuming Need lots of groups of participants Still cannot make causal claims Cohort-sequential design

57. Small N designs What are they? Historically, these were the typical kind of design used until 1920’s when there was a shift to using larger sample sizes Even today, in some sub-areas, using small N designs is common place (e.g., psychophysics, clinical settings, expertise, etc.)

58. Small N designs One or a few participants Data are typically not analyzed statistically; rather rely on visual interpretation of the data Observations begin in the absence of treatment (BASELINE) Then treatment is implemented and changes in frequency, magnitude, or intensity of behavior are recorded

59. Small N designs Baseline experiments – the basic idea is to show: 1. when the IV occurs, you get the effect 2. when the IV doesn’t occur, you don’t get the effect (reversibility)  Before introducing treatment (IV), baseline needs to be stable  Measure level and trend

60. Small N designs Level – how frequent (how intense) is behavior? Are all the data points high or low? Trend – does behavior seem to increase (or decrease) Are data points “flat” or on a slope?

61. ABA design ABA design (baseline, treatment, baseline) – The reversibility is necessary, otherwise something else may have caused the effect other than the IV (e.g., history, maturation, etc.)

62. Small N designs Advantages Focus on individual performance, not fooled by group averaging effects Focus is on big effects (small effects typically can’t be seen without using large groups) Avoid some ethical problems – e.g., with non- treatments Allows to look at unusual (and rare) types of subjects (e.g., case studies of amnesics, experts vs. novices) Often used to supplement large N studies, with more observations on fewer subjects

63. Small N designs Disadvantages Effects may be small relative to variability of situation so NEED more observation Some effects are by definition between subjects Treatment leads to a lasting change, so you don’t get reversals Difficult to determine how generalizable the effects are

64. Small N designs Some researchers have argued that Small N designs are the best way to go. The goal of psychology is to describe behavior of an individual Looking at data collapsed over groups “looks” in the wrong place Need to look at the data at the level of the individual