Term | Description |
---|---|
mean of sample i | |
S^{2}_{i} | variance of sample i |
X_{ij} | j^{th} measurement of the i^{th} sample |
n_{i} | size of sample i |
When n_{1} = n_{2} , the test statistic is Z^{2}. If the null hypothesis, ρ = ρ_{0} is true, then Z^{2} is distributed as a chi-square distribution with 1 degree of freedom. Z^{2} is given by:
where se(ρ_{0}) is the standard error of the pooled kurtosis, which is given by:
where r_{i} = ( n_{i} - 3) / n_{i} and is the pooled kurtosis, which is given by:
se^{2}(ρ_{0}) can also be expressed in terms of the kurtosis values of the individual samples, , as follows:
where:
Let z^{2} be the value of Z^{2} that is obtained from the data. Under the null hypothesis, H_{0}: ρ = ρ_{0} , Z is distributed as the standard normal distribution. Therefore, the p-values for the alternative hypotheses (H_{1}) are given by the following.
Hypothesis | P-value |
---|---|
H_{1}: ρ_{0} ≠ ρ_{0} | P = 2P(Z > |z|) |
H_{1}: ρ_{0} > ρ_{0} | P = P(Z > z) |
H_{1}: ρ_{0} < ρ_{0} | P = P(Z < z) |
Term | Description |
---|---|
S_{i} | the standard deviation of sample i |
ρ | the ratio of the population standard deviations |
ρ_{0} | the hypothesized ratio of the population standard deviations |
α | the significance level for the test = 1 - (the confidence level / 100) |
n_{i} | the number of observations in sample i |
the kurtosis value for sample i | |
X_{ij} | the j^{th} observation in sample i |
m_{i} | the trimmed mean for sample i with trim proportions of |
When n_{1} ≠ n_{2} , there is no test statistic. Rather, the p-value is calculated by inverting the confidence interval procedure. The p-value for the test is given by:
P = 2 min (α_{L}, α_{U})
where c_{α} is an equalizer constant described below and se(ρ_{0}) is the standard error of the pooled kurtosis, which is given by:
where r_{i} = (n_{i} - 3) / n_{i} and is the pooled kurtosis which is given by:
se(ρ_{0}) can also be expressed in terms of the kurtosis values of the individual samples. For more information go to Test for Bonett's method with balanced designs.
The constant vanishes when the designs are balanced, and its effect becomes negligible with increasing sample sizes.
Finding α_{L} and α_{U} is equivalent to finding the roots of the functions L(z , n_{1} , n_{2} , S_{1} , S_{2} ) and L(z , n_{2} , n_{1} , S_{2} , S_{1} ), where L(z , n_{1} , n_{2} , S_{1} , S_{2}) is given by:
To calculate α_{U}, apply the previous steps using the function, L(z, n_{2}, n_{1}, S_{2}, S_{1}), instead of the function, L(z, n_{1}, n_{2}, S_{1}, S_{2}).
Term | Description |
---|---|
S_{i} | the standard deviation of sample i |
ρ | the ratio of the population standard deviations |
ρ_{0} | the hypothesized ratio of the population standard deviations |
α | the significance level for the test = 1 - (the confidence level / 100) |
z_{α} | the upper α percentile point of the standard normal distribution |
n_{i} | the number of observations in sample i |
X_{ij} | the j^{th} observation in sample i |
m_{i} | the trimmed mean for sample i with trim proportions of |
where c_{α/2} is an equalizer constant (described below) and se(ρ) is the standard error of the pooled kurtosis (described below). Typically, this equation has two solutions, a solution, L < S_{1} / S_{2}, and a solution U > S_{1} / S_{2}. L is the lower confidence limit, and U is the upper confidence limit. For more information, go to Bonett's Method, which is a white paper that has simulations and other information about Bonett's Method.
The confidence limits for the ratio of the variances are obtained by squaring the confidence limits for the ratio of the standard deviations.
The constant vanishes when the designs are balanced, and its effect becomes negligible with increasing sample sizes.
se(ρ) is the standard error of the pooled kurtosis which is given by:
where r_{i} = (n_{i} - 3) / n_{i} and is the pooled kurtosis which is given by:
se(ρ) can also be expressed in terms of the kurtosis values of the individual samples. For more information, see the section on the Test for Bonett's method with balanced designs.
Term | Description |
---|---|
α | the significance level for the test = 1 - (the confidence level / 100) |
S_{i} | the standard deviation of sample i |
ρ | the ratio of the population standard deviations |
z_{α/2} | the upper α/2 percentile point of the standard normal distribution |
n_{i} | the number of observations in sample i |
X_{ij} | the j^{th} observation in sample i |
m_{i} | the trimmed mean for sample i with trim proportions of |
Levene’s test is appropriate for continuous data. Levene’s test is not available for summarized data.
To test the null hypothesis that σ_{1} / σ_{2} = ρ with Levene’s test, Minitab performs a one-way ANOVA on the values Z_{1j} and ρZ_{2j} (where j = 1, …, n_{1} or n_{2}).
The Levene's test statistic equals the value of the F-statistic in the resulting ANOVA table. The Levene's test p-value equals the p-value in this ANOVA table.
Under the null hypothesis, the test statistic follows an F-distribution with degrees of freedom DF1 and DF2.
DF1 = 1
DF2 = n_{1} + n_{2} – 2
Term | Description | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Z_{ij} | |X_{i j} – η _{i}|
| ||||||||||
σ_{1} | standard deviation of the first population | ||||||||||
σ_{2} | standard deviation of the second population | ||||||||||
n_{1} | size of the first sample | ||||||||||
n_{2} | size of the second sample |
For continuous data, Minitab calculates the confidence limits for the ratio (ρ) between the population standard deviations with the following formulas. To obtain limits for the ratio between population variances, square the values below.
If , lower limit =
If , a lower limit does not exist
If , upper limit =
If , an upper limit does not exist
If , then
If , then
Term | Description |
---|---|
t _{ α} | the α critical value of a t distribution with n_{1} + n_{2} – 2 degrees of freedom |
η_{i} | the median of sample i |
Z_{ij} | where j = 1, 2, ... , n_{i} and i = 1, 2, and X_{ij} are individual observations |
M_{i} | the mean of Z_{ij} |
S_{i}^{2} | the sample variance of Z_{ij} |
v_{i} | |
ρ | σ_{1} / σ_{2} |
n_{1} | the size of the first sample |
n_{2} | the size of the second sample |
The F-test is appropriate for normal data. To test the null hypothesis that σ_{1} / σ_{2} = ρ with the F-test, Minitab uses the following formulas.
Under the null hypothesis, the F-statistic follows an F-distribution with degrees of freedom DF1 and DF2.
DF1 = n_{1} – 1
DF2 = n_{2} – 1
Term | Description |
---|---|
ρ | σ_{1} / σ_{2} |
σ_{1} | standard deviation of the first population |
σ_{2} | standard deviation of the second population |
S^{2}_{1} | variance of the first sample |
S^{2}_{2} | variance of the second sample |
n_{1} | size of the first sample |
n_{2} | size of the second sample |
When the data follow a normal distribution, Minitab calculates the confidence bounds for the ratio (ρ) between the population standard deviations with the following formulas. To obtain bounds for the ratio between population variances, square the values below.
When you specify a "not equal" alternative hypothesis, a 100(1 – α)% confidence interval for ρ is given by:
When you specify a "less than" alternative hypothesis, a 100(1 – α)% upper confidence bound for ρ is given by:
When you specify a "greater than" alternative hypothesis, a 100(1 – α)% lower confidence bound for ρ is given by:
Term | Description |
---|---|
S_{1} | standard deviation of the first sample |
S_{2} | standard deviation of the second sample |
ρ | σ_{1} / σ_{2} |
n_{1} | size of the first sample |
n_{2} | size of the second sample |
F_{(α/2, n2–1, n1–1)} | α/2 critical value from the F-distribution with degrees of freedom n_{2}–1 and n_{1}–1. |