1. Chapter 13: Comparing Several Means (One-Way ANOVA)
November 18

2. In Chapter 13: 13.1 Descriptive Statistics
11/16/2018
In Chapter 13:
13.1 Descriptive Statistics
13.2 The Problem of Multiple Comparisons
13.3 Analysis of Variance
13.4 Post Hoc Comparisons
13.5 The Equal Variance Assumption
13.6 Introduction to Nonparametric Tests
Basic Biostat

3. Illustrative Example: Data
Pets as moderators of a stress response. This chapter follows the analysis of data from a study in which heart rates (bpm) of participants were monitored after being exposed to a psychological stressor. Participants were randomized to one of three groups:
Group 1 - monitored in presence of pet dog
Group 2 - monitored in the presence of human friend
Group 3 - monitored with neither dog nor human friend present

4. Illustrative Example: Data

5. SPSS Data Table Most computer programs require data in two columns
One column is for the explanatory variable (group)
One column is for the response variable (hrt_rate)

6. 13.1 Descriptive Statistics
Data are described and explored before moving to inferential calculations
Here are summary statistics by group:

7. Exploring Group Differences
John Tukey taught us the importance of exploratory data analysis (EDA)
EDA techniques that apply:
Stemplots
Boxplots
Dotplots
John W. Tukey
( )

8. Side-by-Side Stemplots

9. Side-by-Side Boxplots

10. §13.2 The Problem of Multiple Comparisons
Consider a comparison of three groups. There are three possible t tests when considering three groups:
(1) H0: μ1 = μ2 versus Ha: μ1 ≠ μ2
(2) H0: μ1 = μ3 versus Ha: μ1 ≠ μ3
(3) H0: μ2 = μ3 versus Ha: μ2 ≠ μ3
However, we do not perform separate t tests without modification → this would identify too many random differences

11. Problem of Multiple Comparisons
Family-wise error rate = probability of at least one false rejection of H0
Assume three null hypotheses are true:
At α = 0.05, the Pr(retain all three H0s) = (1−0.05)3 =
Therefore, Pr(reject at least one) = 1−0.847 =  this is the family-wise error rate.
The family-wise error rate is much greater than intended. This is “The Problem of Multiple Comparisons”

12. Problem of Multiple Comparisons
The more comparisons you make, the greater the family-wise error rate. This table demonstrates the magnitude of the problem

13. Mitigating the Problem of Multiple Comparisons
Two-step approach:
1. Test for overall significance using a technique called “Analysis of Variance”
2. Do post hoc comparison on individual groups

14. 13.3 Analysis of Variance One-way ANalysis Of VAriance (ANOVA)
Categorical explanatory variable
Quantitative response variable
Test group means for a significant difference
Statistical hypotheses
H0: μ1 = μ2 = … = μk
Ha: at least one of the μis differ
Method: compare variability between groups to variability within groups (F statistic)

15. Analysis of Variance Overview, cont.
R. A. Fisher
( )
The F in the F statistic stands for “Fisher”

16. Variability Between Groups
Variability of group means around the grand mean → provides a “signal” of group difference
Based on a statistic called the Mean Square Between (MSB)
Notation
SSB ≡ sum of squares between
dfB ≡ degrees of freedom between
k ≡ number of groups
x-bar ≡ grand mean
x-bari ≡ mean of group i

17. Mean Square Between: Formula
Sum of Squares Between [Groups]
Degrees of Freedom Between
Mean Square Between

18. Mean Square Between: Graphically

19. Mean Square Between: Example

20. Variability Within Groups
Variability of data points within groups → quantifies random “noise”
Based on a statistic called the Mean Square Within (MSW)
Notation
SSW ≡ sum of squares within
dfW ≡ degrees of freedom within
N ≡ sample size, all groups combined
ni ≡ sample size, group I
s2i ≡ variance of group i

21. Mean Square Within: Formula
Sum of Squares Within
Degrees of Freedom Within

22. Mean Square Within: Graphically

23. Mean Square Within: Example

24. The F statistic and ANOVA table
Data are arranged to form an ANOVA table
F statistic is the ratio of the MSB to MSW
Fstat “signal-to-noise” ratio

25. Fstat and P-value
The Fstat has numerator and denominator degrees of freedom: df1 and df2 respectively (corresponding to dfB and dfW)
Convert Fstat to P-value with a computer program or Table D
The P-value corresponds to the area in the right tail beyond

26. (Table D does not have df2 of 42; next lowest df2 is 30).
Table D (“F Table”)
The F table has limited listings for df2.
You often must round-down to the next available df2 (rounding down preferable for conservative estimate).
Wedge the Fstat between listing to find the approximate P-value
df1 = 2, df2 = 42
(Table D does not have df2 of 42; next lowest df2 is 30).
Fstat = falls below P of 0.001

27. Fstat and P-value
P < 0.001

28. ANOVA Example (Summary)
Hypotheses: H0: μ1 = μ2 = μ3 vs. Ha: at least one of the μis differ
Statistics: Fstat = with 2 and 42 degrees of freedom
P-value = (via SPSS), providing highly significant evidence against the H0; conclude the heart rates (an indicator of the effects of stress) differed in the groups
Significance level (optional): Results are significantly at α =

29. Computation
Because of the complexity of computations, ANOVA statistics are often calculated by computer

30. ANOVA and the t test (Optional)
ANOVA for two groups is equivalent to the equal variance (pooled) t test (§12.4)
Both address H0: μ1 = μ2
dfW = df for t test = N – 2
MSW = s2pooled
Fstat = (tstat)2
F1,df2,α = (tdf,1-α/2)2

31. 13.4 Post Hoc Comparisons
ANOVA Ha says “at least one population mean differs” but does not delineate which differ.
Post hoc comparisons are pursued after rejection of the ANOVA H0 to delineate differences

32. SPSS Post Hoc Comparison Procedures
Many post hoc comparison procedures exist. We cover the LSD and Bonferroni methods.

33. Least Squares Difference Procedure
Do after a significant ANOVA to protect against the problem of multiple comparisons
Hypotheses. H0: μi = μj vs. Ha: μi ≠ μj for each group i and j
Test statistic
C. P-value. Use t table or software

34. LSD Procedure: Example
For the “pets” illustrative data, we test H0: μ1 = μ2 by hand. The other tests will be done by computer
A. Hypotheses. H0: μ1 = μ2 against Ha: μ1 ≠ μ2
B. Test statistic.
C. P-value. P = ; highly significant evidence of a difference.

35. LSD Procedure, SPSS Results for illustrative “pets” data. H0: μ1 = μ2

36. 95% Confidence Interval, Mean Difference, LSD Method

37. 95% CI, LSD Method, Example
Comparing Group 1 to Group 2:

38. Bonferroni Procedure
The Bonferroni procedure is instituted by multiplying the P-value from the LSD procedure by the number of post hoc comparisons “c”.
Hypotheses. H0: μ1 = μ2 against Ha: μ1 ≠ μ2
Test statistic. Same as for the LSD method.
P-value. The LSD method produced P = (two-tailed). Since there were three post hoc comparisons, PBonf = 3 × =

39. Bonferroni Confidence Interval
Let c represent the number of post hoc comparisons. Comparing Group 1 to Group 2:

40. Bonferroni Procedure, SPSS
P-values from Bonferroni are higher and confidence intervals are broader than LSD method, reflecting its conservative approach

41. §13.5. The Equal Variance Assumption
Conditions for ANOVA:
1. Sampling independence
2. Normal sampling distributions of mean
3. Equal variance within population groups
Let us focus on condition 3, since conditions 1 and 2 are covered elsewhere.
Equal variance is called homoscedasticity. (Unequal variance = heteroscedasticity).
Homoscedasticity allows us to pool group variances to form the MSW

42. Assessing “Equal Variance”
1. Graphical exploration. Compare spreads visually with side-by-side plots.
2. Descriptive statistics. If a group’s standard deviation is more than twice that of another, be alerted to possible heteroscedasticity
3. Test variances. A statistical test can be applied (next slide).

43. Levene’s Test of Variances
Hypotheses. H0: σ21 = σ22 = … = σ2k Ha: at least one σ2i differs
B. Test statistic. Test is performed by computer. The test statistic is a particular type of Fstat based on the rank transformed deviations (see p. 283 for details).
C. P-value. The Fstat is converted to a P-value by the computational program. Interpretation of P is routine  small P  evidence against H0, suggesting heteroscedasticity.

44. Levene’s Test – Example (“pets” data)
A. H0: σ21 = σ22 = σ23 versus Ha: at least one σ2i differs
B. SPSS output (below). Fstat = with 2 and 42 df
C. P = Very weak evidence against H0  retain assumption of homoscedasticity

45. Analyzing Groups with Unequal Variance
Stay descriptive. Use summary statistics and EDA methods to compare groups.
Remove outliers, if appropriate (p. 287).
Mathematically transform the data to compensate for heteroscedasticity (e.g., a long right tail can be pulled in with a log transform).
Use robust non-parametric methods.

46. 13.6 Intro to Nonparametric Methods
Many nonparametric procedures are based on rank transformed data (“rank tests”). Here are examples:

47. The Kruskal-Wallis Test
Let us explore the Kruskal-Wallis test as an example of a non-parametric test
The Kruskal-Wallis test is the non-parametric analogue of one-way ANOVA.
It does not require Normality or Equal Variance conditions for inference.
It is based on rank transformed data and seeing if the mean ranks in groups differ significantly.

48. Kruskal-Wallis Test
The K-W hypothesis can be stated in terms of mean or median (depending on assumptions made about population shapes). Let us use the later.
Let Mi ≡ the median of population i
There are k groups
H0: M1 = M2 = … = Mk
Ha: at least one Mi differs

49. Kruskal-Wallis, Example
Alcohol and income. Data from a survey on alcohol consumption and income are presented.

50. Kruskal-Wallis Test, Example
We wish to test whether the means differ significantly but find graphical and hypothesis testing evidence that the population variances are unequal.

51. Kruskal-Wallis Test, Example, cont.
Hypotheses. H0: M1 = M2 = M3 = M4 = M5 vs. Ha: at least one Mi differs
Test statistic. Some computer programs use chi-square statistic based upon a Normal approximation. SPSS derives Chi-square statistics = with 4 df (next slide)

52. Kruskal-Wallis Test, Example, cont.
P = 0.099, providing marginally significant evidence against H0.