Testing variable found in mutually exclusive groups from same population, normally distributed or not, can show if groups are equivalent {analysis of variance} {variance analysis} (ANOVA).
measurements
In ANOVA, measurements are sums of four parts: mean, class or treatment effect, sampling or measurement effect, and normally distributed random error. ANOVA checks if sampling error or class error is great enough compared to random error to make samples or classes actually different.
process
Assume groups are equivalent and so have same mean. Set significance level to 5%.
For two groups, calculate variance ratio {variance ratio} {F value, ANOVA}. Group degrees of freedom are group number minus one: C - 1. Sample degrees of freedom df are sum of group degrees of freedom Ni, minus group number C. df = N1 + N2 + ... - C.
Numerator is sample-mean-difference variance: ( (sum from i = 1 to i = N1 of (n1(i))^2) / N1 - (sum from i = 1 to i = N1 + N2 + ... of (n(i))^2) / (N1 + N2 + ...) ) / (C - 1) + ( (sum from i = 1 to i = N2 of (n2(i))^2) / N2 - (sum from i = 1 to i = N1 + N2 + ... of (n(i))^2) / (N1 + N2 + ...) ) / (C - 1) + ... Denominator is population variance: ( (sum from i = 1 to i = N1 + N2 + ... of (n(i))^2) - ( (sum from i = 1 to i = N1 of (n1(i))^2) / N1 + (sum from i = 1 to i = N2 of (n2(i))^2) / N2 + ... ) ) / (N1 + N2 + ... - C), where n is frequency, N is sample size, and C is sample number.
F values form distributions {F distribution, ANOVA} that vary with degrees of freedom and significance.
If calculated F value is less than F value in F distribution for same degrees of freedom and significance level, accept that samples have same mean. If calculated F value is more, reject hypothesis, so at least one sample is not random, or samples are from different populations.
mean square
For samples or treatments, degrees of freedom can divide into sum of squares of differences between values and mean {mean square}. Mean square estimates population variance. F value is sample or treatment mean square divided by error mean square.
missing data
Least-squares method can estimate missing data.
types
Replications are like classes {randomized blocks plan}.
Testing interactions between treatments can be at same time as testing interactions between samples {two-way classification}.
comparison to t test
With two samples, F test and t test are similar.
Mathematical Sciences>Statistics>Tests
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Date Modified: 2022.0224