EXAMPLE 1 Butter Fat Content in Cow Milk

7 - One-way Analysis of Variance (ANOVA)

Suppose we wish to compare k population means (). This situation can arise in two ways. If the study is observational, we are obtaining independently drawn samples from k distinct populations and we wish to compare the population means for some numerical response of interest. If the study is experimental, then we are using a completely randomized design to obtain our data from k distinct treatment groups. In a completely randomized design the experimental units are randomly assigned to one of k treatments and the response value from each unit is obtained. The mean of the numerical response of interest is then compared across the different treatment groups.

There two main questions of interest:

1) Are there at least two population means that differ?

2) If so, which population means differ and how much do they differ by?

More formally:

If we reject the null then we use comparative methods to answer question 2 above.

Assumptions:

1. Samples are drawn independently (completely randomized design)

2. Population variances are equal, i.e. .

3. Populations are normally distributed.

We examine checking these assumptions in the examples that follow.

The test procedure compares the variation in observations between samples to the variation within samples. If the variation between samples is large relative to the variation within samples we are likely to conclude that the population means are not all equal. The diagrams below illustrate this idea...

Between Group Variation > Within Group Variation Between Group Variation Within Group Variation

(Conclude population means differ) (Fail to conclude the population means differ)

Hypothetical Example: Comparing k = 9 Treatments

Within Group Variation

To measure the variation within groups we use the following formula:

This is an estimate of the variance common to all k populations ().

Between Group Variation

To measure the variation between groups we use the following formula:

The larger the differences between the k sample means for the treatment groups are, the larger will be.

If the null hypothesis is true, i.e. the k treatments have the same population mean, we expect the between group variation to the same as the within group variation (). If we have evidence against the null hypothesis in support of the alternative, i.e. we have evidence to suggest that at least two of the population means are different.

Test Statistic

~ F-distribution with numerator df = k -1 and denominator df = n – k

The larger F is the strong the evidence against the null hypothesis, i.e. Big FàReject Ho.

EXAMPLE 1 - Weight Gain in Anorexia Patients
Data File: Anorexia.JMP
These data give the pre- and post-weights of patients being treated for anorexia nervosa. There are actually three different treatment plans being used in this study, and we wish to compare their performance in terms of the mean weight gain of the patients. The patients in this study were randomly assigned to one of three therapies.
The variables in the data file are:

group - treatment group (1 = Family therapy, 2 = Standard therapy, 3 = Cognitive Behavioral therapy)

Treatment – treatment group by name (Behavioral, Family, Standard)

prewt - weight at the beginning of treatment

postwt- weight at the end of the study period

Weight Gain - weight gained (or lost) during treatment (postwt-prewt)

We begin our analysis by examining comparative displays for the weight gained across the three treatment methods. To do this select Fit Y by X from the Analyze menu and place the grouping variable, Treatment, in the X box and place the response, Weight Gain, in the Y box and click OK. Here boxplots, mean diamonds, normal quantile plots, and comparison circles have been added.

Things to consider from this graphical display:

· Do there appear to be differences in the mean weight gain?

· Are the weight changes normally distributed?

· Is the variation in weight gain equal across therapies?

Checking the Equality of Variance Assumption

To test whether it is reasonable to assume the population variances are equal for these three therapies select UnEqual Variances from the Oneway Analysis pull down-menu.

We have no evidence to conclude that the variances/standard deviations of the weight gains for the different treatment programs differ (p > .05).

ONE-WAY ANOVA TEST FOR COMPARING THE THERAPY MEANS
To test the null hypothesis that the mean weight gain is the same for each of the therapy methods we will perform the standard one-way ANOVA test. To do this in JMP select Means, Anova/t-Test from the Oneway Analysis pull-down menu. The results of the test are shown in the Analysis of Variance box.

The p-value contained in the ANOVA table is .0065, thus we reject the null hypothesis at the .05 level and conclude statistically significant differences in the mean weight gain experienced by patients in the different therapy groups exist.
MULTIPLE COMPARISONS

Because we have concluded that the mean weight gains across treatment method are not all equal it is natural to ask the secondary question:

Which means are significantly different from one another?

We could consider performing a series of two-sample t-Tests and constructing confidence intervals for independent samples to compare all possible pairs of means, however if the number of treatment groups is large we will almost certainly find two treatment means as being significantly different. Why? Consider a situation where we have k = 7 different treatments that we wish to compare. To compare all possible pairs of means (1 vs. 2, 1 vs. 3, …, 6 vs. 7) would require performing a total of two-sample t-Tests. If we used for each test we expect to make Type I Error, i.e. we expect to find one pair of means as being significantly different when in fact they are not. This problem only becomes worse as the number of groups, k , gets larger.

Experiment-wise Error Rate

Another way to think about this is to consider the probability of making no Type I Errors when making our pair-wise comparisons. When k = 7 for example, the probability of making no Type I Errors is , i.e. the probability that we make at least one Type I Error is therefore .6596 or a 65.96% chance. Certainly this unacceptable! Why would conduct a statistical analysis when you know that you have a 66% of making an error in your conclusions? This probability is called the experiment-wise error rate.

Bonferroni Correction

There are several different ways to control the experiment-wise error rate. One of the easiest ways to control experiment-wise error rate is use the Bonferroni Correction. If we plan on making m comparisons or conducting m significance tests the Bonferroni Correction is to simply use as our significance level rather than . This simple correction guarantees that our experiment-wise error rate will be no larger than . This correction implies that our p-values will have to be less than rather than to be considered statistically significant.

Multiple Comparison Procedures for Pair-wise Comparisons of k Population Means

When performing pair-wise comparison of population means in ANOVA there are several different methods that can be employed. These methods depend on the types of pair-wise comparisons we wish to perform. The different types available in JMP are summarized briefly below:

· Compare each pair using the usual two-sample t-Test for independent samples. This choice does not provide any experiment-wise error rate protection! (DON’T USE)

· Compare all pairs using Tukey’s Honest Significant Difference (HSD) approach. This is best choice if you are interested comparing each possible pair of treatments.

· Compare with the means to the “Best” using Hsu’s method. The best mean can either be the minimum (if smaller is better for the response) or maximum (if bigger is better for the response).

· Compare each mean to a control group using Dunnett’s method. Compares each treatment mean to a control group only. You must identify the control group in JMP by clicking on an observation in your comparative plot corresponding to the control group before selecting this option.

Multiple Comparison Options in JMP

EXAMPLE 1 - Weight Gain in Anorexia Patients (cont’d)

For these data we are probably interested in comparing each of the treatments to one another. For this we will use Tukey’s multiple comparison procedure for comparing all pairs of population means. Select Compare Means from the Oneway Analysis menu and highlight the All Pairs, Tukey HSD option. Beside the graph you will now notice there are circles plotted. There is one circle for each group and each circle is centered at the mean for the corresponding group. The size of the circles are inversely proportional to the sample size, thus larger circles will drawn for groups with smaller sample sizes. These circles are called comparison circles and can be used to see which pairs of means are significantly different from each other. To do this, click on the circle for one of the treatments. Notice that the treatment group selected will be appear in the plot window and the circle will become red & bold. The means that are significantly different from the treatment group selected will have circles that are gray. These color differences will also be conveyed in the group labels on the horizontal axis. In the plot below we have selected Standard treatment group.

The results of the pair-wise comparisons are also contained the output window.

The matrix labeled Comparison for all pairs using Tukey-Kramer HSD identifies pairs of means that are significantly different using positive entries in this matrix. Here we see only treatments 2 and 3 significantly differ.

The next table conveys the same information by using different letters to represent populations that have significantly different means. Notice treatments 2 and 3 are not connected by the same letter so they are significantly different.

Finally the CI’s in the Ordered Differences section give estimates for the differences in the population means. Here we see that mean weight gain for patients in treatment 3 is estimated to be between 2.09 lbs. and 13.34 lbs. larger than the mean weight gain for patients receiving treatment 2 (see highlighted section below).

EXAMPLE 2 – Motor Skills in Children and Lead Exposure

These data come from a study of children living in close proximity to a lead smelter in El Paso, TX. A study was conducted in 1972-73 where children living near the lead smelter were sampled and their blood lead levels were measured in both 1972 and in1973. IQ tests and motor skills tests were conducted on these children as well. In this example we will examine difference in finger-wrist tap scores for three groups of children. The groups were determined using their blood lead levels from 1972 and 1973 as follows:

· Lead Group 1 = the children in this group had blood levels below 40 micrograms/dl in both 1972 and 1973 (we can think of these children as a “control group”).

· Lead Group 2 = the children in this group had lead levels above 40 micrograms/dl in 1973 (we can think of these children as the “currently exposed” group).

· Lead Group 3 = the children in this group had lead levels above 40 microgram/dl in 1972, but had levels below 40 in 1973 (we can think of these children as the “previously exposed” group).

The response that was measured (MAXFWT) was the maximum finger-wrist tapping score for the child using both their left and right hands to do the tapping (to remove hand dominance). These data are contained in the JMP file: Maxfwt Lead.JMP.

Select Fit Y by X and place Lead Group in the X,Factor box and MAXFWT in the Y,Response box. After selecting Quantiles and Normal Quantile Plots we obtain the following comparative display.

The wrist-tap scores appear to be approximately normally distributed. The variance in the wrist-tap scores may appear to be a bit larger for Lead Group 1, however because this is the largest group we expect to see more observations in the extremes. The interquartile ranges for the three lead groups appear to be approximately equal. We can check the equality of variance assumption formally by selecting the UnEqual Variance option.

Formally Checking the Equality of the Population Variances

ONE-WAY ANOVA FOR COMPARING THE MEAN WRIST-TAP SCORES ACROSS LEAD GROUP

To test the null hypothesis that the mean finger wrist-tap score is the same for each of the lead exposure groups we will perform the standard one-way ANOVA test. To do this in JMP select Means, Anova from the Oneway Analysis pull-down menu. The results of the test are shown in the Analysis of Variance box.

The p-value contained in the ANOVA table is .0125, thus we reject the null hypothesis at the .05 level and conclude that statistically significant differences in the mean finger wrist tap scores of the children in the different lead exposure groups exist (p < .05).

Multiple Comparisons using Tukey’s HSD

Tukey’s HSD Pair-wise Comparisons (cont’d)

The results of the pair-wise comparisons are also contained the output window.

The matrix labeled Comparison for all pairs using Tukey-Kramer HSD identifies pairs of means that are significantly different using positive entries in this matrix. Here we see only lead groups 1 and 2 significantly differ.