1. Statistics Psych 231: Research Methods in Psychology

2. Reminders Quiz 9 (chapters 12 & 13) is due on Nov. 5 th (tonight) at midnight Journal Summary 2 assignment Due in class next week (Wednesday, Nov. 12 th ) 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.

3. Statistics Mistrust of statistics? It is all in how you use them They are a critical tool in research

4. Samples and Populations Sample Population Sampling methods

5. Samples and Populations 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? Sample Inferential statistics used to generalize back Population

6. Samples and Populations 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? Sample Inferential statistics used to generalize back Population

7. Distribution Recall that a variable is a characteristic that can take different values. The distribution of a variable is a summary of all the different values of a variable Both type (each value) and token (each instance ) How much do you like statistics? 1 - 2 - 3 - 4 - 5 Hate it Love it How much do you like statistics? 1 - 2 - 3 - 4 - 5 Hate it Love it 5 values (1, 2, 3, 4, 5) 155413 7 tokens (1,1,2,3,4,5,5) 2

8. Distribution Many important distributions Population All the scores of interest Sample All of the scores observed (your data) Used to estimate population characteristics Distribution of sample distributions Used to estimate sampling error Population Sample 512 1 3 5 3 1 1 2 3 1 2 1 5 21 3 5 1 3 3 5 2 How do we describe these distributions? Use descriptive statistics, focus on 3 properties

9. Describing Distributions 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

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

11. Mean & Standard deviation 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): Add up all of the X’s Divide by the total number in the population Divide by the total number in the sample mean

12. Mean & Standard deviation 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 The mean (mathematical average) is the most popular and most important measure of center. Others include median and mode.

13. 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) standard deviation = σ =

14. 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 An Example: Computing Standard Deviation (sample) 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)

15. 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? Sample Inferential statistics used to generalize back Population

16. 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? Population Sample A Treatment X = 80% Sample B No Treatment X = 76%

17. 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

18. 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)

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

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

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

28. 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

29. 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

30. 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”

31. 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

32. 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

33. 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

34. “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

35. “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”

36. “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

37. “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

38. Sampling error 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”

39. “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

40. “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

41. “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”)

42. “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

43. 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

44. 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

45. 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

46. 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

47. 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.”

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

55. 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