QUANT homework

ANOVA

ANOVA

• Analysis of Variance • Statistical method to analyzes variances to determine if the means from more than

two populations are the same • compare the between-sample-variation to the within-sample-variation • If the between-sample-variation is sufficiently large compared to the within-sample-

variation it is likely that the population means are statistically different

• Compares means (group differences) among levels of factors. No assumptions are made regarding how the factors are related

• Residual related assumptions are the same as with simple regression

• Explanatory variables can be qualitative or quantitative but are categorized for group investigations. These variables are often referred to as factors with levels (category levels)

ANOVA Assumptions

• Assume populations , from which the response values for the groups are drawn, are normally distributed

• Assumes populations have equal variances • Can compare the ratio of smallest and largest sample standard deviations.

Between .05 and 2 are typically not considered evidence of a violation assumption

• Assumes the response data are independent

• For large sample sizes, or for factor level sample sizes that are equal, the ANOVA test is robust to assumption violations of normality and unequal variances

ANOVA and Variance

Fixed or Random Factors

• A factor is fixed if its levels are chosen before the ANOVA investigation begins • Difference in groups are only investigated for the specific pre-selected factors

and levels

• A factor is random if its levels are choosen randomly from the population before the ANOVA investigation begins

Randomization

• Assigning subjects to treatment groups or treatments to subjects randomly reduces the chance of bias selecting results

ANOVA hypotheses statements

One-way ANOVA

One-Way ANOVA Hypotheses statements

Test statistic

= 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒

𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒

Under the null hypothesis both the between and within group variances estimate the variance of the random error so the ratio is assumed to be close to 1.

Null Hypothesis

Alternate Hypothesis

One-Way ANOVA

One-Way ANOVA

One-Way ANOVA Excel Output

Treatment groups

Total 0.391495833 23 p-value is found using

the F-statistic and the F-

distribution.

The p-value for this test is less than 0.05. Reject the null hypothesis and conclude that at least one treatment group mean is statistically different.

Multiple Comparison Tests If the ANOVA test of the null hypothesis is rejected, then conclude that not all the means are equal but doesn’t suggest which means are statistically different

Multiple Comparison Test

One-Way ANOVA Example problem

One-Way ANOVA Example problem

One-Way ANOVA Example problem

Studentized Range q table

One-Way ANOVA Example problem