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One-Way Analysis of Variance or One-Way ANOVA is a statistical method to determine if there is a difference in means between two or more independent groups, where the groups are defined by the outcomes for a single categorical variable.
In its purpose, it is essentially an extension of the independent samples test for a difference in means, extended to more than two groups.
Suppose there are \(K\geq 2\) number of groups. The null hypothesis says that the means of all the groups are equal to each other. The alternative hypothesis says that at least one of the means differs from another mean.
Null: \(\mu_1 = \mu_2 = ... = \mu_K\)
Alternative: \(\mu_i \neq \mu_j\) for some \(i\neq j\)
The following assumptions are necessary for the one-way ANOVA test:
Random assignment: The groups the observations belong to are determined independently of the outcome variable.
Independent groups: Observations in one group are independent of observations in all other groups. There must therefore be different sampling units in each group.
Homogeneity of variance: The population variances of the outcome variable for the different groups are the same, even though the population means may be different.
Central Limit Theorem assumptions: There is a sufficiently large sample size and/or the population distribution for the outcome variable is normal.
The data set below includes data from more than 52,000 individuals over the age of 25 years that participated in the 2016 Current Population Survey. The variables include the following:
HOURS: Usual weekly hours worked over all jobs (ratio)SEX: Male or female (categorical)RACE: Racial identity (categorical)EDUC: Education (categorical)MARST: Marital status (categorical)The following code downloads the data, opens it, and saves it in a data frame called wdat.
load(url("http://murraylax.org/datasets/cpshours.RData"))We can use this data to determine if the number of usual weekly hours is different for people in different categories for one or more of the categorical variables.
The lm() function can be used to estimate several types of linear models including a one-way or multi-way ANOVA. The code below estimates a one-way ANOVA that predicts usual weekly hours worked by race.
lmhours <- lm(HOURS ~ 0 + RACE , data=wdat)The first parameter is a formula that tells lm() to estimate a linear relationship that predicts the outcome variable HOURS with the explanatory variable RACE. Because RACE is a categorical variable, lm() will treat each category within RACE as its own explanatory variable.
Those who are familiar with using the lm() function to estimate a regression will find this code familiar. A possible difference is the inclusion of 0 + in the formula preceding the explanatory variable RACE. This is entered in the formula so that lm() does not estimate an intercept coefficient, but rather estimates the unique intercept (which is equal to the mean in a one-way ANOVA) for each category in RACE.
We can see the estimates for the mean number of categories for each race by investigating the coefficients.
lmhours$coefficients## RACEAmerican Indian/Aleut/Eskimo RACEAsian/Pacific Islander
## 39.44427 40.40267
## RACEBlack RACEOther
## 40.08817 40.73537
## RACEWhite
## 40.64012The coefficients are equal to the mean of each category. This implies the mean usual weekly work hours for American Indian / Aleut / Eskimo people is 39.4, the mean usual work hours for Black people is 40.1, etc.
The code below reports the 95% confidence intervals for these means.
confint(lmhours, level=0.95)## 2.5 % 97.5 %
## RACEAmerican Indian/Aleut/Eskimo 38.58930 40.29925
## RACEAsian/Pacific Islander 40.05854 40.74680
## RACEBlack 39.79435 40.38199
## RACEOther 39.97124 41.49949
## RACEWhite 40.53238 40.74787The ANOVA procedure uses a decomposition of variances to determine how much variability in the outcome variable is explained by different assignments to the categorical variable, and how much variability is unexplained.
The Between Group Variation is a measure of explained variation, the measure of variability in the outcome variable that is explained by assignment to different categories of the explanatory variable. In our example, it is the measure of variability in usual weekly hours that is explained by a person being in a given racial category.
The Within Groups Variation is a measure of unexplained variation, also called residual variation. It is the measure of variability in the outcome variable that is seen within each category of the explanatory variable. In our example, it is the variability in usual weekly hours between individuals within any single racial group.
When between groups variation is sufficiently large compared to within groups variation, there is evidence to reject the null hypothesis and conclude there is sufficient statistical evidence that the outcome variable has a different population mean for the different categories of the explanatory variable.
We can call the anova() function to conduct the ANOVA test. This will estimate each of these measures of variability and determine if average usual weekly hours is different for people of different races.
anova(lmhours)## Analysis of Variance Table
##
## Response: HOURS
## Df Sum Sq Mean Sq F value Pr(>F)
## RACE 5 86008324 17201665 138020 < 2.2e-16 ***
## Residuals 52301 6518380 125
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We pass in a single parameter to the anova() function, the return object from the lm() function. We can see here that the p-value is very low (less than \(2.2 \times 10^{-16}\)). We reject the null hypothesis and conclude that the population mean usual weekly hours is different for different racial groups.
Having established that mean usual weekly hours is different for people of different races, we can take the analysis a step further and make pairwise comparisons of the means between different races. We can do this by running a series of independent-samples t-tests for each pair of races. Since there are 5 racial categories, there are 10 different pairs, and therefore 10 different independent samples t-tests to run.
Remember that any hypothesis test can lead to incorrectly rejecting the null hypothesis by statistical chance. This is called a Type 1 error. The probability of making a type 1 error is equal to the significance level you use for the hypothesis test, usually 5%.
When we run 10 independent-samples t-tests, the probability of making a type 1 error in at least one of the 10 tests is larger. For example, if the chances of making a type 1 error are independent across the 10 statistical tests and each equal to 5%, then probability of making a type 1 error in at least one of the tests is equal to \(1 - (1 - 0.05)^{10} = 0.4013\), or more than 40%!
The Bonferroni post-hoc tests are a series of independent-samples t-tests for differences in means that makes a correction for p-values so that the probability of making one or more type 1 errors over all the difference tests is not more than 5%.
The function call pairwise.t.test() that follows estimates independent-samples t-tests for differences in means tests for each pair of race categories, making a Bonferroni corrections to the p-values.
pairwise.t.test(wdat$HOURS, wdat$RACE, p.adjust.method="bonf")##
## Pairwise comparisons using t tests with pooled SD
##
## data: wdat$HOURS and wdat$RACE
##
## American Indian/Aleut/Eskimo Asian/Pacific Islander
## Asian/Pacific Islander 0.4154 -
## Black 1.0000 1.0000
## Other 0.2733 1.0000
## White 0.0653 1.0000
## Black Other
## Asian/Pacific Islander - -
## Black - -
## Other 1.0000 -
## White 0.0055 1.0000
##
## P value adjustment method: bonferroniThe only racial pair where the corrected p-value is less than 5% is White people vs. Black / African American people, where the p-value is equal to 0.0055. We find sufficient statistical evidence that there is a difference in usual weekly hours between these two groups.