Williams Test For The Minimum Effective Dose
Williams Test For The Minimum Effective Dose
Williams Test For The Minimum Effective Dose
Chapter 595
Test Procedure
This section will summarize the Williams’ test procedure. Those needing more details should look
at Williams (1971, 1972).
Suppose that a set of treatments are comprised of a control (treatment 0) and K dose levels of
substance of interest (treatments i = 1, …, K). All treatment groups have the same sample size n.
Assume that the effect of the substance is to increase the mean response and that the mean response
is proportional to the size of the dose. That is, assume that
μ0 ≤ μ1 ≤ μ2 ≤ L ≤ μ K .
The statistical hypothesis that is tested by the Williams’ test is
H 0 : μ0 = μ1 = L = μK vs. H a : μ0 = μ1 = L = μi −1 < μi ≤ μi +1 ≤ L ≤ μK .
The test statistic is given by
μˆi − X 0
Ti =
1 1
s +
ni n0
where s is an unbiased estimate of σ , the within group standard deviation that is statistically
independent of the X i , and μ̂i is the maximum likelihood estimate of μi given by
595-2 Williams Test for the Minimum Effective Dose
⎧ v ⎫
⎪∑nj X j ⎪
⎪ ⎪
μˆi = max min ⎨ j = uv ⎬
1≤ u ≤ i i ≤ v ≤ K
⎪ ∑nj ⎪
⎪⎩ j = u ⎪⎭
The hull hypothesis of no treatment difference is rejected and the fact that the ith dose level is the
minimum effective dose is concluded if
T j > t j ,α for all j ≥ i
where t j ,α is the upper αth percentile of the distribution of T j . These critical values were
tabulated by Williams (1972) and they are available in PASS using special interpolation routines.
The only values of α that are available are 0.05, 0.025, 0.01, and 0.005.
Technical Details
Computing Power
The following approximate function for power is given by Chow et al. (2008) page 288.
⎛ Δ ⎞
1 − β = 1 − Φ ⎜ tK ,α − ⎟
⎝ σ 2/n ⎠
z
where Δ is the clinically meaningful minimal difference and Φ (z ) = ∫ Normal (0,1) .
−∞
2σ 2 [tK ,α + zβ ]
2
n= .
Δ2
Procedure Options
This section describes the options that are specific to this procedure. These are located on the
Data tab. For more information about the options of other tabs, go to the Procedure Window
chapter.
Data Tab
The Data tab contains most of the parameters and options that you will be concerned with.
Williams Test for the Minimum Effective Dose 595-3
Solve For
Find (Solve For)
This option specifies the parameter to be calculated from the values of the other parameters.
Under most conditions, you would select either Power and Beta or n.
Select n when you want to determine the sample size needed to achieve a given power and alpha
error level.
Select Power and Beta when you want to calculate the power of an experiment that has already
been run.
Error Rates
Power or Beta
This option specifies one or more values for power or for beta (depending on the chosen setting).
Power is the probability of rejecting a false null hypothesis, and is equal to one minus beta. Beta
is the probability of a type-II error, which occurs when a false null hypothesis is not rejected.
Values must be between zero and one. Historically, the value of 0.80 (beta = 0.20) was used for
power. Now, 0.90 (beta = 0.10) is also commonly used.
A single value may be entered here or a range of values such as 0.8 to 0.95 by 0.05 may be
entered.
Alpha (Significance Level)
This option specifies a value for the probability of a type-I error. A type-I error occurs when a
true null hypothesis is rejected.
Only four values are available: 0.05, 0.025, 0.01, 0.005.
Sample Size
G (Number of Groups)
Select the number of groups in the study (including the control group). Since the experiment
compares K doses to a single control (zero dose), there are actually K+1 groups. Values from 3 to
11 are available.
For example, suppose an experiment includes three doses and a control group, you would enter
‘4’ as the total number of groups.
n (Sample Size per Group)
Enter one or more values for the group sample size (n). This is the number of individuals in each
group. The Williams' design assumes that the number of subjects in each group is constant. The
total number of subjects is N = n(G).
You may enter a list of values such as
10 20 30 40 50
or
10 to 100 by 10.
595-4 Williams Test for the Minimum Effective Dose
Setup
This section presents the values of each of the parameters needed to run this example. First, from
the PASS Home window, load the Williams’ Test for the Minimum Effective Dose procedure
window by expanding Means, then ANOVA, then clicking on Multiple Comparisons, and then
clicking on Williams’ Test for the Minimum Effective Dose. You may then make the
appropriate entries as listed below, or open Example 1 by going to the File menu and choosing
Open Example Template.
Williams Test for the Minimum Effective Dose 595-5
Option Value
Data Tab
Find (Solve For) ...................................... n (Sample Size)
Power ...................................................... 0.90
Alpha ....................................................... 0.05
G (Number of Groups) ............................ 5
n (Sample Size per Group) ..................... Ignored since this is the search parameter.
∆ (Minimum Detectable Difference) ........ 10 15 20 30 40 50
σ (Standard Deviation) ............................ 25
Annotated Output
Click the Run button to perform the calculations and generate the following output.
Numeric Results
Numeric Results for Williams' Test
Null Hypothesis: μ[0] = μ[1] = ... = μ[K]
Alternative Hypothesis: μ[0] = μ[1] = ... = μ[i-1] < μ[i] ≤ μ[i+1] ≤ ... ≤ μ[K]
References
Chow, S.C.; Shao, J.; Wang, H. 2008. Sample Size Calculations in Clinical Research, Second Edition. Chapman &
Hall. New York. Pages 287 - 293.
Williams, D.A. 1971. 'A Test for Differences between Treatment Means When Several Dose Levels are Compared
with a Zero Dose Control', Biometrics, Volume 27, No. 1, pages 103-117.
Williams, D.A. 1972. 'The Comparison of Several Dose Levels with a Zero Dose Control', Biometrics, Volume 28,
No. 1, pages 519-531.
Report Definitions
Power is the probability of rejecting a false null hypothesis.
G is the number of groups.
n is the sample size of each dose level group.
N is the total sample size which is equal to n(G).
Alpha is the probability of rejecting a true null hypothesis.
Beta is the probability of accepting a false null hypothesis.
∆ (delta) is the minimum difference between a treatment mean and the control mean that is to be detected by
the study.
σ (sigma) is the standard deviation within each group.
Summary Statements
A total sample size of 580 (or 116 in each of the 5 groups) achieves 90% power to detect a
minimal difference of 10.00 between the zero-dose mean and a treatment mean with a significance
level (alpha) of 0.0500 and a standard deviation of 25.00 using Williams' Test.
This report shows the values of each of the parameters, one scenario per row. The values of
power and beta were calculated from the other parameters.
595-6 Williams Test for the Minimum Effective Dose
Power
Power is the probability of rejecting a false null hypothesis.
G
G is the number of groups in the study including a zero-dose control group.
n
n is the sample size of each dose level group.
N
N is the total sample size which is equal to n(G).
Alpha
Alpha is the probability of rejecting a true null hypothesis.
Beta
Beta is the probability of accepting a false null hypothesis.
∆ (Minimum Detectable Difference)
∆ (delta) is the minimum difference between a treatment mean and the control mean that is to be
detected by the study.
σ (Standard Deviation)
σ (sigma) is the standard deviation within each group.
Plots Section
This plot shows the relationship between the group sample size and the minimal difference.
Williams Test for the Minimum Effective Dose 595-7
Setup
This section presents the values of each of the parameters needed to run this example. First, from
the PASS Home window, load the Williams’ Test for the Minimum Effective Dose procedure
window by expanding Means, then ANOVA, then clicking on Multiple Comparisons, and then
clicking on Williams’ Test for the Minimum Effective Dose. You may then make the
appropriate entries as listed below, or open Example 2 by going to the File menu and choosing
Open Example Template.
Option Value
Data Tab
Find (Solve For) ...................................... n (Sample Size)
Power ...................................................... 0.80
Alpha ....................................................... 0.05
G (Number of Groups) ............................ 4
n (Sample Size per Group) ..................... Ignored since this is the search parameter.
∆ (Minimum Detectable Difference) ........ 11
σ (Standard Deviation) ............................ 22
Annotated Output
Click the Run button to perform the calculations and generate the following output.
Numeric Results
Numeric Results for Williams' Test
Null Hypothesis: μ[0] = μ[1] = ... = μ[K]
Alternative Hypothesis: μ[0] = μ[1] = ... = μ[i-1] < μ[i] ≤ μ[i+1] ≤ ... ≤ μ[K]
Note that PASS has rounded up to the next integer, while Chow rounded down. Thus, n = 54
guarantees a power of at least 80%.
595-8 Williams Test for the Minimum Effective Dose