class4medsci.udel.edu/open/statclass/january2011/class4.pdf · title: microsoft powerpoint -...

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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


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


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.


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

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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).


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


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.



• 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.

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Data Characterizations



Data Characterizations

SPSS Summary output

Mean RBP4 increases with obesity (bmi)



Data CharacterizationsGraphical Investigation: Side by Side Boxplots



• 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)

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(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



(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

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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.


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.


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

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Multiple comparisons (SPSS demo)


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


Kruskal-Wallis Test: Data Analysis


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.

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Thank you

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