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Unformatted text preview: Statistical Methods I (EXST 7005) Page 123 We could also express our null hypothesis in terms of EMS [ H0: στ = 0 ], particularly for the
random effect since the variance component for treatments may be a value of interest.
2 Since for a fixed effect the individual means are usually of interest, the null hypothesis is
usually expressed in terms of the means ( H0: μ1 = μ2 = μ3 = ... = μt ). Descriptions of post-hoc tests
Post-hoc or Post-ANOVA tests! Once you have found out some treatment(s) are “different”,
how do you determine which one(s) are different? If we had done a t-test on the individual pairs of treatments, the test would have been done as
Y1 − Y2
Y1 − Y2
. If the difference between Y1 − Y2 was large
enough, the t value would have been greater than the tcritical and we would conclude that
there was a significant difference between the means. Since we know the value of tcritical
we could figure out how large a difference is needed for significance for any particular
values of MSE, n1 and n2. We do this by replacing t with tcritical and solving for Y1 − Y2 . ( t= ( ) Y1 −Y2
n1 n2 ( tcritical MSE ) ) Y1 −Y2
n1 n2 = ( ( n1 + n1 ) = Y − Y
1 1 2 2 ) , so or Y1 − Y2 = tcritical SY1 −Y2 This value is the exact width of an interval Y1 − Y2 which would give a t-test equal to tcritical. Any
larger values would be “significant” and any smaller values would not. This is called the
“Least Significant Difference”. LSD = tcritical SY −Y 1 2 This least significant difference calculation can be used to either do pairwise tests on observed
differences or to place a confidence interval on observed differences.
The LSD can be done in SAS in one of two ways. The MEANS statement produces a range
test (LINES option) or confidence intervals (CLDIFF option), while the LSMEANS
statement gives pairwise comparisons.
The LSD has an α probability of error on each and every test. The whole idea of ANOVA is to
give a probability of error that is α for the whole experiment, so, much work in statistics
has been dedicated to this problem. Some of the most common and popular alternatives
are discussed below. Most of these are also discussed in your textbook. The LSD is the LEAST conservative of those discussed, meaning it is the one most likely
to detect a difference and it is also the one most likely to make a Type I error when it finds
a difference. However, since it is unlikely to miss a difference that is real, it is also the
most powerful. The probability distribution used to produce the LSD is the t distribution. James P. Geaghan Copyright 2010 Statistical Methods I (EXST 7005) Page 124 Bonferroni's adjustment. Bonferroni pointed out that in doing k tests, each at a probability of
Type I error equal to α, the overall experimentwise probability of Type I error will be NO
MORE than k*α, where k is the number of tests. Therefore, if we do 7 tests, each at
α=0.05, the overall rate of error will be NO MORE than = 0.35, or 35%. So, if we want to
do 7 tests and keep an error rate of 5% overall, we can do each individual test at a rate of
α/k = 0.055/7 = 0.007143. For the 7 tests we have an overall rate of 7*0.007143 = 0.05.
The probability distribution used to produce the LSD is the t distribution.
Duncan's multiple range test. This test is intended to give groupings of means that are not
significantly different among themselves. The error rate is for each group, and has
sometimes been called a familywise error rate. This is done in a manner similar to
Bonferroni, except the calculation used to calculate the error rate is [1-(1-α)r-1] instead of
the sum of α. For comparing two means that are r steps apart, where for adjacent means
r=2. Two means separated by 3 other means would have r = 5, and the error rate would be
[1-(1-α)r-1] = [1-(1-0.05)4] = 0.1855. The value of a needed to keep an error rate of α is the
reverse of this calculation, [1-(1-0.05)1/4] = 0.0127.
Tukey's adjustment The Tukey adjustment allows for all possible pairwise tests, which is
often what an investigator wants to do. Tukey developed his own tables (see Appendix
table A.7 in your book for “percentage points of the studentized range”). For “t”
treatments and a given error degrees of freedom the table will provide 5% and 1% error
rates that give an experimentwise rate of Type I error.
Scheffé's adjustment This test is the most conservative. It allows the investigator to do not
only all pairwise tests, but all possible tests, and still maintain an experimentwise error
rate of α. “All possible” tests includes not only all pairwise tests, but comparisons of all
possible combinations of treatments with other combinations of treatments (see
CONTRASTS below). The calculation is based on a square root of the F distribution, and
can be used for range type tests or confidence intervals. The test is more general than the
others mentioned, for the special case of pairwise comparisons, the statistic is √(t–1)*Ft-1,
n(t-1) for a balanced design with t treatments and n observations per treatment. Place the post-hoc tests above in order from the one most likely to detect a difference (and the
one most likely to be wrong) to the one least likely to detect a difference (and the one least
likely to be wrong). LSD is first, followed by Duncan's test, Tukey's and finally
Scheffé's. Dunnett's is a special test that is similar to Tukey's, but for a specific purpose,
so it does not fit well in the ranking. The Bonferroni approach produces an upper bound
on the error rate, so it is conservative for a given number of tests. It is a useful approach if
you want to do a few tests, fewer than allowed by one of the others (e.g. you may want to
do just a few and not all possible pairwise). In this case, the Bonferroni may be better.
Evaluating the assumptions for ANOVA. We have already discussed some techniques for the evaluation of data for homogeneous
variance. The assumption of independence is somewhat more difficult to evaluate.
Random sampling is the best guarantee of independence and should be used as much as
The third assumption is normality. The observations are assumed to be normally
distributed within each treatment, but how the treatments come together to form the
dependent variable Yij may cause them to look non-normal. The best way to test for
normality is to examine the residuals, pooling the normal distribution across the
James P. Geaghan Copyright 2010 Statistical Methods I (EXST 7005) Page 125 treatments to a common mean of zero. SAS will output the residuals with an output
statement, and PROC UNIVARIATE has a number of tools to evaluate normality.
Homogeniety of Variance Your textbook discusses one test by Hartley. It is one of the simplest tests, but not usually
the best. To do this test we calculate the largest observed variance divided by the
smallest observed variance. This statistics is tested with a special table by Hartley
(Appendix Table 5.A in your Freund & Wilson textbook).
A number of other tests are available in SAS, but only for a simple CRD (i. e. a One-way
ANOVA). These test are briefly discussed below.
To get all of the tests available in SAS, use the following statement following PROC
MEANS your_treatment_name / HOVTEST=BARTLETT
HOVTEST=LEVENE(TYPE=SQUARE) HOVTEST=OBRIEN WELCH; Levene's Test: This test is basically an ANOVA of the squared deviations
(TYPE=SQUARE). It can also be done with absolute values (TYPE=ABS). This is
one of the most popular HOV tests.
O'Brien's Test: This test is a modification of Levene's with an additional adjustment for
Brown and Forsythe's Test: This test is similar to Levene's, but uses absolute deviations
from the median instead of more ANOVA like means. There is a “nonparametric”
ANOVA that employs deviations from the median instead of the usual deviations
from the mean used for the normal ANOVA.
Bartlett's Test for Equality: This test is similar to Hartley's, but uses a likelihood ratio
test instead of an F test. This test can be inaccurate if the data is not normally
Welch's ANOVA: It is not a test of homogeneity of variance; this test is a weighted
ANOVA. This ANOVA weights the observations by an inverse function of the
variances and is intended to address the problem of non-homogeneous variance and
to be use when the variance is not homogeneous.
The Homogeniety of Variance (HOV) tests discussed above can be done in SAS (PROC
GLM). Note that the last one is NOT an HOV test, it is another type of ANOVA
called a weighted ANOVA. Contrasts and Orthogonality
A priori contrasts are one of the most useful and powerful techniques in ANOVA. There are a few
additional considerations that should be made. So what is a contrast? As described in the handout, it is a comparison of some means against
some other means. The comparison is a linear combination.
When we set these up in SAS, we only need to give the multipliers in the CORRECT ORDER,
and SAS will complete the calculations.
The multipliers must sum to zero, and they can be given as fractions or as integers. James P. Geaghan Copyright 2010 ...
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- Fall '08