1
Nemours Biomedical Research
Statistics
February 16, 2011
Jobayer Hossain, Ph.D., Tim Bunnell, Ph.D. &
Larry Holmes, Ph.D.
Hypothesis Testing: More than two groups
Nemours Biomedical Research
Class Objectives -- You will Learn:
• Hypothesis testing of quantitative response variables involving more
than two groups (samples) using Parametric and Non-parametric
methods.
• Parametric method: One way Analysis of variance (ANOVA) to
compare means of three or more groups
• Non-parametric method: Kruskal Wallis test for comparing medians
(mean of rank) of three or more groups
• Multiple comparisons
• Performing One way ANOVA and Kruskal Wallis test using SPSS
Nemours Biomedical Research
Analysis of variance (ANOVA)
• The analysis of variance (ANOVA) is a technique of decomposing
the total variability of a response variable into:
• Variability due to the experimental factor(s) and…
• Variability due to error (i.e., factors that are not accounted for in
the experimental design).
• The basic purpose of ANOVA is to test the equality of several means.
Nemours Biomedical Research
One-way ANOVA
� The basic ANOVA situation
�Type of variables: Quantitative response and Categorical (factor)
predictors (independent variable).
� Main Question: Are mean response measures of different groups
equal?
� One categorical predictor with only 2 levels (groups):
� 2-sample t-test
� One categorical predictor with more than two levels (groups):
� One way ANOVA
� Two or more categorical predictors, each with at least two or more
levels (groups) of each:
� Factorial ANOVA
2
Nemours Biomedical Research
One-way ANOVA
• Assumptions:
– Observations are independent.
– The response variable is normally distributed in
each group.
– Standard deviations (SD) for all groups are
approximately equal (rule of thumb: ratio of largest
to smallest are approximately 2:1).
Nemours Biomedical Research
One-way ANOVA
• Hypothesis:Ho: Means of all groups are equal.
Ha: At least one of them is not equal to other.
– doesn’t say how or which ones differ.
– Can be followed up with “multiple comparisons”
• ANOVA Table for one way classified data
n-1SSTTotal
MSE=SSE/n-kn-kSSEError
F=MSG/MSEMSG=SSG/k-1k-1SSGGroup
F-RatioMean Sum of Squares
dfSum of Squares
Sources of Variation
Note: Large F means that MSG is large compared to MSE
Nemours Biomedical Research
One-way ANOVA: Data Analysis
• We will use the second dataset (“Retinoid”) in the class website
for the SPSS demonstration.
• The dataset contains few variables from a clinical trial-
“EFFECTS OF A FRUIT AND VEGETABLE JUICE
CONCENTRATE (FVJC) IN-VIVO ON RETINOL BINDING
PROTEIN 4 AND ANTIOXIDANT CAPACITY IN NORMAL AND
OVERWEIGHT BOYS –A RANDOMIZED PLACEBO
CONTROLLED STUDY”
• J. ATILIO CANAS, MD is the Principal Investigator of this study.
Nemours Biomedical Research
One-way ANOVA: Data Analysis
• Suppose, we want to compare the average baseline
Retinol Binding Protien 4 (variable RBP40M in the
dataset) between lean, overweight and obese
(variable bmigrp3 in the dataset) patients.
• We will characterize data, check assumptions and
will use One-way ANOVA to compare the mean
baseline RBP4 between three groups if assumptions
are met.
3
Nemours Biomedical Research
One-way ANOVA: Data Analysis
Data Characterizations
Nemours Biomedical Research
One-way ANOVA: Data Analysis
Data Characterizations
SPSS Summary output
Mean RBP4 increases with obesity (bmi)
Nemours Biomedical Research
One-way ANOVA: Data Analysis
Data CharacterizationsGraphical Investigation: Side by Side Boxplots
Nemours Biomedical Research
One-way ANOVA: Data Analysis
• Checking Assumptions:
• Normality of the response variable:
• Shapiro-Wilk test: A p-value of less than or equal to .05 indicates a significant
deviation from Normality.
• Normal Q-Q plot: If plotted points of the response variable scattered around
approximately a diagonal line, we can assume the normality of the distribution
• SPSS Demo: Analyze -> Descriptive Statistics -> Explore -> In the Explore
window, bring the response variable under the dependent list and the group
(categorical) variable under the Factor list and then select plots from this
window. It will open a small window- Explore Plots. Select Normality Plots
with tests from this window (Demo in the next slide)
4
Nemours Biomedical Research
One-way ANOVA: Data Analysis
(checking Normality)
Nemours Biomedical Research
One-way ANOVA: Data Analysis
(checking Normality)
Nemours Biomedical Research
One-way ANOVA: Data Analysis
(checking Normality)
P-value (Sig. In the output) of Shapiro-Wilk test is greater than .05 for each group. It simply indicates no violation of the assumption of Normality.
SPSS output: Shapiro-Wilk test
Nemours Biomedical Research
One-way ANOVA: Data Analysis
(checking Normality: Normal Q-Q plot)
Points of RBP40 falls approximately on the diagonal line. We can assume the normality of this variable
for lean group
5
Nemours Biomedical Research
One-way ANOVA: Data Analysis
Nemours Biomedical Research
One-way ANOVA: SPSS Output
Levene test for homogeneity of variances between
groups:p-value ≤.05 indicates the non-equality of variances
between groups
P-value (Sig. in the output) =.846 indicates the
homogeneity of variances between groups.
Nemours Biomedical Research
One-way ANOVA: SPSS Output and
Interpretation
P-value (Sig. in the output) of F statistic is .081. The mean
baseline RBP4 between three groups are not significantly different (p>.05).
In case of p-value < .05, at least for one group, the mean baseline RBP4 is significantly different from other groups. In that case, we need to perform pairwise (multiple)
comparisons of groups to identify the group (s) significantly different from other groups.
Nemours Biomedical Research
Multiple comparisons
• If the F test is significant in ANOVA table, then we
intend to find the pairs of groups are significantly
different. Following are the commonly used
procedures:– Fisher’s Least Significant Difference (LSD)
– Tukey’s HSD method
– Bonferroni’s method
– Scheffe’s method
– Dunn’s multiple-comparison procedure
– Dunnett’s Procedure
6
Nemours Biomedical Research
Multiple comparisons (SPSS demo)
Nemours Biomedical Research
Kruskal-Wallis Test
• A distribution free non-parametric test to
compare the equality of median between
three or more groups.
• No distributional assumption is needed.
• Identical to the one-way ANOVA with the data
replaced by their ranks.
• An extension of Mann-Whitney U test
Nemours Biomedical Research
Kruskal-Wallis Test: Data Analysis
Nemours Biomedical Research
Kruskal-Wallis Test: Data AnalysisSPSS Output and Interpretation
The p-value (Asymp. Sig.) is 0.146. There is no significant difference in median between three groups.
7
Nemours Biomedical Research
Thank you