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Index to Module 4 Notes
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In Module Notes 4.1 we
discussed methods designed to compare means of two independent samples
to determine if the means of the populations from which they were drawn are
equal or not. In Module Notes 4.2 we presented a method for comparing means of
two related samples to determine if the means of the populations from
which they were drawn are equal or not. Finally, Module Notes 4.3 provided a
nonparametric method of comparing two independent samples to determine
of the medians of the populations from which they were drawn are equal or not,
when the assumption of normal population and equal variances could not be met.
We are now going to expand the concept presented in Module Notes 4.1 to compare
means of multiple samples (beyond two) to determine if the means of the
populations from which they were drawn are equal or not. Suppose we were
interested in comparing means of three samples to determine if the means of the
populations from which they were drawn are equal or not. The hypotheses
statements are:
H0:
MeanA = MeanB = MeanC
Ha: At least two Means are not equal
The parametric technique we use
to conduct the analysis is called Analysis of Variance (ANOVA). The title
Analysis of Variance may imply that we are going to compare variances, and not
means. Actually we are going to compare means, but the way we do it is to
compare the difference or variation between the means of the three
groups to the variation within the groups. If the variation between the
means is greater than the variation within the groups, we go with the
alternative hypothesis.
Formulas for computing between group variation and within group variation are
available in the references at the end of this set of module notes. We are
going to let Excel do the computations. The important thing to remember is that
the between group variation simply measures how the sample means compare to the
grand average of all of the data. Does at least one group have a sample mean
much greater or much less than the grand average of all the data? If
"yes" (meaning the difference between the means is significantly
greater than the within group variation), then we reject the null hypothesis
and conclude at least two of the means are not equal. The within group
variation I am referring to measures how the individual observations within a
group differ from their group sample mean.
When the sample means are far from each other relative to the within group
variation, "something" is going on. In the miles per gallon scenario,
the brands of gasoline really have different properties that shifts the average
mpg of one or more groups away from other another group or groups. Simply put,
the average mpg is significantly different between the groups. When the sample
means are close together relative to the within group variation, we say that
any difference between sample means is due to chance - nothing "special"
is going on to cause the means to be different.
You may recall that we already studied an ANOVA table - we did that in
regression. In that case, we compared the variation attributed to the
regression model to the variation that was unexplained (the error or residual).
We are going to use the ANOVA table again, this time comparing between group
variation to within group variation.
The
Situation
I will return to the
miles per gallon study introduced in Module Notes 4.1. Here is the data:
Worksheet 4.4.1
|
Brand A |
Brand B |
Brand C |
|
20 |
20 |
20 |
|
20 |
20.5 |
26 |
|
19 |
18.5 |
23 |
|
16 |
20 |
24 |
|
15 |
19 |
23 |
|
17 |
19 |
25 |
|
14 |
18 |
23 |
The objective is to
compare the mean mpg of group A, with group B with group C to determine if there
is or is not a difference between the means. The hypotheses statements are
presented in the introduction above. We are going to use the F statistic
because we are comparing the ratio of two variances (between group variation to
within group variation).
The tool we use in Excel is a Data Analysis add in. So, we select Tools
from the Standard Toolbar, Data Analysis from the pulldown menu, then ANOVA
Single Factor. (In Excel 2007 you have to first select Data, then select Data
Analysis, then ANOVA Single Factor).
Then respond to the dialog box questions. Note that my data is in
columns, and I defaulted to an alpha of 0.05. Single Factor means we are
studying one factor that may result in the means being different - that factor
is the type or brand of gasoline put into the 7 cars in each of the three
groups. The results (output) of the ANOVA analysis are shown below:
Worksheet 4.4.2
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Anova: Single Factor |
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SUMMARY |
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Groups |
Count |
Sum |
Average |
Variance |
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Brand A |
7 |
121 |
17.28571429 |
5.904761905 |
||
|
Brand B |
7 |
135 |
19.28571 |
0.821429 |
||
|
Brand C |
7 |
164 |
23.42857143 |
3.619047619 |
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ANOVA |
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|
Source of Variation |
SS |
df |
MS |
F |
P-value |
F crit |
|
Between Groups |
137.4286 |
2 |
68.71429 |
19.92635 |
2.73237E-05 |
3.554561 |
|
Within Groups |
62.07143 |
18 |
3.448413 |
|||
|
Total |
199.5 |
20 |
The first thing to note in
the ANOVA output are the descriptive statistics which include the means and the
variances of the three samples. These are followed by the ANOVA Table which has
the variation separated as Between Groups (variation between the three means
and the grand mean) and Within Groups. Since the p-value of 2.732E-05 is less
than the alpha value of 0.05, we reject the null hypothesis and conclude that
there is a difference between the means. Thus, "something" special is
going on - the brands of gas do have a significant differential effect on the
mpg performance.
Now that we have some statistics, we can look at the between and within group
variation with numbers. Note that Group A's sample mean is 17.3, and Group C's
sample mean is 23.4. The difference between these two groups is 6.1 mpg,
a measure of the between group variation. The variation within Group A
is 2.4 (measured by the standard deviation of Group A which is just the square
root of Group A variance) and within group C it's 1.9. Using the larger of the
two, we see that the difference between the groups of 6.1 is much larger
than the largest of the two variations within the groups, hence we
rejected the null hypothesis... "something" is going on to cause the
difference other than chance alone. In the actual computation, the ANOVA
technique computes and uses a pooled variance as the measure of variation
within the group, so the above was done simply to illustrate the concept.
Note carefully the alternative hypothesis simply says there is a difference
between the means somewhere, but at this point, we don't know where that is -
we suspect it's between A and C as demonstrated above. To find the answer to
the question, "which pairs of means are different," we do a post
hoc test (post hoc meaning a follow-on test after the ANOVA). We would only
do this when we reject the null hypothesis, since whenever we fail to reject
the null hypothesis in ANOVA, we are saying there is no difference between the
means.
The Bonferroni Multiple Comparisons Procedure
The Bonferroni Procedure (Sincich, Business Statistics by Example)
uses information from the ANOVA table, and a t-Test statistic, so it can be
done in Excel. There are other post hoc tests, such as the Tukey Cramer
procedure (Levine, 1999), but these require special tables not available in
Excel. I believe the Bonferroni has higher utility to the manager interested in
using the statistical capability of Excel.
The first thing to do is determine how many pairs of means we want to test for
differences. Since there are three groups, we can do a maximum of three tests:
compare the mean of A with B, the mean of A with C and the mean of B with C.
The second thing to do is determine the difference between the sample means:
Eq.
4.4.1: | MeanA - MeanB | = | 17.3 - 19.3 | = 2
Eq. 4.4.2: | MeanA - MeanC | = | 17.3 - 23.4 | = 6.1
Eq. 4.4.3: | MeanB - MeanC | = | 19.3 - 23.4 | = 4.1
where | | means absolute value.
The third task is to compute
the Bonferroni critical difference statistic, B. This statistic becomes the
threshold value for comparison. Any difference between sample means (such as
those shown in Equations 4.4.1 - 4.4.3) greater than B is a statistically
significant difference - those two means are not equal. Any difference between
sample means less than B is not a significant difference - those two means are
equal.
Equation 4.4.4 provides the formula for B:
Eq. 4.4.4: B = t (alpha/2g)* Sq Root (MS Within Groups) * Sq Root [(1/n1) + (1/n2)]
where: g = number of pair wise comparisons being made. Note, if there are three groups, the maximum number of comparisons is 3*(3-1)/2 = 3; if there are five groups, the maximum is 5*(5-1)/2 = 10; n1 and n2 are the sample sizes of the two groups being compared.
The t(alpha/2g)
can be obtained from Excel. Let's let alpha be 0.05, g is 3, so alpha/2*3 would
be 0.00833. To get a t-score for 0.00833 with 18 degrees of freedom associated
with the Within Group variation (see ANOVA Table in Worksheet 4.4.2 above) we
use the =TINV(alpha, degrees of freedom) function in Excel. In an active cell
in an Excel Worksheet, select Insert on the Standard Toolbar, Function
from the pulldown menu, Statistical, TINV, and enter 0.00833
for alpha and 18 for degrees of freedom. (In
Excel 2007 you have to select Formulas, then Insert Function. From the pulldown
menu, select Statistical, TINV).
The function should look like this: =TINV(0.00833, 18) which
gives a t score of 2.963.
The MS Within Groups is the Mean Square Within Groups obtained from the ANOVA
table (3.4484). Now to put it all together:
Eq. 4.4.5: B = 2.963 * Sq Root(3.4484) * Sq Root [(1/7) + (1/7)]
B = 2.941
The Bonferroni Critical
statistic is 2.941. Any pair wise difference between sample means greater than
2.941 is a statistically significant difference. Equation 4.4.2 and 4.4.3 show
that the difference between means of groups A and C and B and C are greater
than 2.941; therefore those pairs of means are significantly different. On the
other hand, the difference between the means of A and B is only 2 (Eq. 4.4.1);
that is not a significant difference, but rather only due to chance.
ANOVA
Assumptions
There are three
assumptions needed to employ the one-way or one factor ANOVA model.
The first assumption is randomness and independence. To avoid bias, it
is important the the data be gathered and placed in the multiple groups in a
random, objective manner so that the observations are independent of each
other. Thus, the one factor ANOVA, also called the One Factor Completely
Randomized Design, does not apply to related samples (such as the same car
being given Brand of Gas A, then B, then C).
The second assumption is the normality assumption: the sampled groups
are drawn from normally distributed populations, just as in the case of
the t-Test for independent samples. The ANOVA procedure works fairly well as
long as there are not extreme departures from normality, especially if the
sample sizes are large. If there are departures from normality, as examined by
looking at descriptive statistics for the samples, and if the sample sizes are
small (less than 30), the nonparametric technique presented in Module Notes
4.4.6 is the preferred alternative.
The last assumption is called homogeneity of variance. This assumptions
requires that the variances of the multiple groups be equal. This is necessary
because the variances of each group are pooled to create the single within
group variation used in comparison to the between group variation in the ANOVA
procedure. If the sample sizes are the same within each group, inferences made
may not be seriously affected. However, if the sample sizes are not the same,
and the variances are unequal, there is a serious effect on the inferences
made.
When both the normality and homogeneity of variance assumptions are violated,
it is best to revert to the nonparametric procedure presented in Module Notes
4.6.
References:
Anderson, D.,
Sweeney, D., & Williams, T. (2010). Essentials of Modern Business
Statistics with Microsoft Excel. Cincinnati, OH: South-Western, Chapter 10 (Sections
10.4 - 10.5).
D. Groebner, P.
Shannon, P. Fry & K. Smith. Business
Statistics: A Decision Making Approach, Fifth Edition, Prentice Hall, Chapter
10
Ken Black. Business Statistics for
Contemporary Decision Making. Fourth Edition, Wiley. Chapter 10
& 11
Levine,
D., Berenson, M. & Stephan, D. (1999). Statistics for Managers Using
Microsoft Excel (2nd. ed.). Upper Saddle River, NJ: Prentice-Hall, Chapter 10.
Mason, R., Lind, D. &
Marchal, W. (1999). Statistical Techniques in Business and Economics (10th.
ed.). Boston: Irwin McGraw Hill, Chapter 11.
Sincich, T. (1992). Business Statistics by Example, (4th. ed.). New
York: Dellen.