Comparison of Multiple Groups

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Comparison of Multiple Groups

• Trey Spencer
• Division of Biometry

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Definitions
A treatment is a specific combination of factor
levels, e.g., 5 ml of Drug A and 10 ml of Drug
B. The response is the variable being measured by
the experimenter, e.g., degree of pain relief.
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Definitions
An experimental unit is the object on which a
measurement (or measurements) is taken. A factor
is an independent variable whose values are
controlled and varied by the experimenter, e.g.,
drug dosage. A level is a specific value of a
factor.
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Analysis of Variance

• In an analysis of variance (ANOVA), the total
  variation in the response measurements is divided
  into portions that may be attributed to various
  factors, e.g., amount of variation due to Drug A
  and amount due to Drug B.
• The variability of the measurements within
  experimental groups and between experimental
  groups and their comparisons is formalized by the
  ANOVA.

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Assumption for anAnalysis of Variance
The observations within each population are
normally distributed. The normal distributions
have a common variance s 2, but may have a
different mean, ?. The observations are
independent.
Analysis of variance procedures are fairly robust
when sample sizes are equal and when the data are
fairly mound-shaped.
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The Completely Randomized Design A One-Way
Classification
A completely randomized design is one in which
random samples are selected independently from
each of k populations.
From this point, the completely randomized design
will be referred to as a one-way ANOVA.
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One-Way ANOVA
Involves only one factor, designated a one-way
classification. The factor has k levels
corresponding to the k populations, which are
also the treatments for this deign. Are the k
population means all the same, or is at least
onedifferent from the others? The ANOVA
compares the population means simultaneously
versus pair by pair as with Students t.
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One-Way ANOVA

• Suppose you want to compare k population means
  based on independent random samples of size n 1,
  n 2, , n K from normal populations with a common
  variance s 2. They have the same shape, but
  potentially different locations

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Partitioning the Total Variance
Total Sums of Squares (Total SS)

• Let xij be the j th measurement ( j 1,2, , ni
  ) in the i th sample. The total sum of squares is

If we let G represent the grand total of all n
observations, then the Total SS can be written as
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Partitioning the Total Variance
Treatment Sums of Squares (SST)

• This Total SS is partitioned into two components.
  The first component, called the sum of squares
  for treatments (SST), measures the variation
  among k sample means

where Ti is the total of the observations for
treatment i.
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Partitioning the Total Variance
Error Sums of Squares (SSE)
The second component, called the sum of squares
for error (SSE), is used to measure the pooled
variations within the k samples
We can show algebraically that, in the analysis
of variance, Total SS SST SSE Therefore,
once you calculate two of the three sums of
squares, the third component can be calculated
easily.
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Degrees of Freedom
Each of the three sources of variance, when
divided by its appropriate degrees of freedom,
provides an estimate of thevariation in the
experiment. Since Total SS involves n squared
observations, its degrees of freedom are d f (n
- 1) . Similarly, the sum of squares for
treatments involves k squared observations, and
its degrees of freedom are d f (k - 1) . The
sum of squares for error has the following df df
(n1-1) (n2-1) (nk-1) n-k . The
degrees of freedom are additive, so that df
(Total) df (TSS) df (SSE)
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Mean Squares
The three sources of variation and their
respective degrees of freedom are combined to
form the mean squares as MS SS/d f. The total
variation in the experiment is then displayed in
an ANOVA table.
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Testing the Equity of the Treatment Means
F test for Comparing k Populations Means 1.
Specify the hypotheses
Ho ?1 ?2 ?k Ha One
or more pairs of population means differ. 2.
Test statistic F MST/ MSE, where F is based on
d f 1 (k - 1) and d f 2 (n - k). 3.
Rejection region Reject H 0 if F gt Fa , where
Fa lies in the upper tail of the F distribution
(with d f 1 k - 1 and d f 2 n - k) or if
p-value lt a. 4. Draw Conclusions.
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Testing the Equity of the Treatment Means

F-Distribution with 3 and 57 degrees of freedom
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Testing the Equity of the Treatment Means

Assumptions The samples are randomly and
independently selected from their respective
populations. The populations are normally
distributed with potentially different means and
equal variances.
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Testing the Equity of the Treatment Means

• Remember that s 2 is the common variance for all
  k populations. The quantity MSE SSE/(n - k) is
  a pooled estimate of s 2, a weighted average of
  all k sample variances, whether or not H 0
• is true.
• If H 0 is true, then the variation in the sample
  means, measured by MST SST/ (k - 1), also
  provides an unbiased estimate of s 2. If H 0 is
  false and the population means are different,
  then MST is unusually large.

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Testing the Equity of the Treatment Means

• The test statistic F MST/ MSE tends to be
  larger that usual if H 0 is false. So, you can
  reject H 0 for large values of F, using a
  right-tailed statistical test.
• Under the assumption that H 0 is true, the test
  statistic has an F distribution with d f 1 (k
  - 1) and d f 2 (n - k) degrees of freedom and
  right-tailed critical values of the F
  distribution can be used.

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ExamplePartial Thromboplastin Time (PTT)

Kohli,et al. Blood Coagulation and Fibrinolysis
2002
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Example

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Example

1. Specify the hypotheses
Ho ?1 ?2 ?3 ?4 Ha
One or more pairs of population means
differ. 2. Test statistic F 15.59, where F
is based on d f 1 3 and d f 2 106. 3.
Rejection region Reject H 0 if F gt F0.05
2.69. 4. Conclusion There is evidence to
suggest that one or more groups differ with
respect to mean PTT.
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Example

P-value Pr(Fgt15.59) 0.0000002
test statistic F15.59
critical value F2.69
We know that at least two groups differ which
ones differ?
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Multiple Comparisons
With multiple comparisons it is useful to define
two kinds of error rates. 1. Comparison-wise
error rate The probability of rejecting the
null hypothesis of no effect for a single
contrast when the null hypothesis is true. 2.
Experiment-wise error rate The probability of
rejecting at least one contrast among a number of
contrasts considered in a given experiment when
the null hypotheses for all contrast are true.
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Multiple Comparisons
If the comparison-wise error rate is ?, the
experiment-wise error rate will be larger than ?
if there are two or more contrasts. Experimentwise
Error 1-(1- ?)r
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Multiple Comparisons

• Two methods for commonly used for controlling the
  experimentwise error
• Bonferroni Method
• Tukeys Method

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Multiple Comparisons
Bonferroni Test

• where
• m Number of comparisons
• s 2 MSE Estimator of the common variance s
  2
• d f Number of degrees of freedom for s 2
• ni Sample size of the groups being compared
  in each of the k treatment means
• t deviate from the t-distribution
• Rule Two population means are judged to differ
  if the corresponding sample means differ by w or
  more.

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Multiple Comparisons
Bonferroni Test
Critical value from t-distribution t106, 0.0042
Many t tables will not provide values for large
df. In those cases, the following formula
provides a good approximation
Estimate Std.Error Y-O 0.63 1.43 Y-E
1.66 1.43 Y-L 8.01 1.43 O-E
1.03 1.28 O-L 7.38 1.28 E-L
6.35 1.28
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Multiple Comparisons
Tukeys Studentized Range Test

• where
• k Number of treatments
• s 2 MSE Estimator of the common variance s
  2
• d f Number of degrees of freedom for s 2
• nt Common sample sizethat is, the number of
  observations in each of the k treatment
  means
• q?(k, df) Tabulated value
• Rule Two population means are judged to differ
  if the corresponding sample means differ by ? or
  more.

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Multiple Comparisons
Tukeys Studentized Range Test
MSE from ANOVA Table
Critical value from Studentized range table
Estimate Std.Error Y-O 0.63 1.43 Y-E
1.66 1.43 Y-L 8.01 1.43 O-E
1.03 1.28 O-L 7.38 1.28 E-L
6.35 1.28
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Multiple Comparisons
Below are other multiple comparison methods that
control the experimentwise error

• All pairwise comparisons
• Sidak Method
• Scheffes Method
• All treatments to a single control
• Dunnetts Method