Test for Equality of Variance

Using var.test() to test equality of variance

• In this procedure, we determine whether or not two populations have the same variance.

• The assumption of equal variance of two populations underpins several inference procedures. This assumption is tested by comparing the variance of samples taken from both populations.

• We will not get into too much detail about that, but concentrate on how to assess equality of variance.

• The null and alternative hypotheses are as follows: \[ H_0: \sigma^2_1 = \sigma^2_2 \] \[ H_1: \sigma^2_1 \neq \sigma^2_2 \]

• The Null hypothesis expounds equality of variance. The alternative hypothesis expounds difference in variance.

• When using R it would be convenient to consider the null and alternative in terms of variance ratios.

• Two data sets have equal variance if the variance ratio is 1.

\[ H_0: \sigma^2_1 / \sigma^2_2 = 1 \] \[ H_1: \sigma^2_1 / \sigma^2_2 \neq 1 \]

x <- c(14,13,16,20,12,18,11,09,13,11)
y <- c(15,13,18,20,10,17,23,11,10)

You would be required to compute the test statistic for this procedure. The test statistic is the ratio of the variances for both data sets. \[ TS = \frac{s^2_x}{s^2_y} \]

The standard deviations would be provided in the following R code.

sd(x); sd(y)

## [1] 3.40098

## [1] 4.630815

• Sample standard deviation for data set \(x\) = 3.40
• Sample standard deviation for data set \(y\) = 4.63

To compute the test statistic. \[ TS = \frac{3.40^2}{4.63^2} = \frac{11.56}{21.43} = 0.5394 \]

Implement the F-Test for Variance

var.test(x,y)

## 
##  F test to compare two variances
## 
## data:  x and y
## F = 0.53938, num df = 9, denom df = 8, p-value = 0.3764
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.1237892 2.2125056
## sample estimates:
## ratio of variances 
##          0.5393782

• The p-value is 0.3764 (top right), above the threshold level of 0.0250, the default threshold in statistical courses. We fail to reject the null hypothesis.

• We can assume that there is no significant difference in sample variances. Therefore we can assume that both populations have equal variance, as there is insufficient evidence to the contrary

• Additionally the \(95\%\) confidence interval (0.1237, 2.2125) contains the condition of equality, i.e. a variance ratio value equal to 1.