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Key Differences Between T-test and ANOVA

The independent groups might be defined by a particular characteristic of the participants such as BMI e. Suppose that the outcome is systolic blood pressure, and we wish to test whether there is a statistically significant difference in mean systolic blood pressures among the four groups. The sample data are organized as follows:. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols.

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The research hypothesis captures any difference in means and includes, for example, the situation where all four means are unequal, where one is different from the other three, where two are different, and so on. The alternative hypothesis, as shown above, capture all possible situations other than equality of all means specified in the null hypothesis. The table can be found in "Other Resources" on the left side of the pages.

Note that N does not refer to a population size, but instead to the total sample size in the analysis the sum of the sample sizes in the comparison groups, e. The test statistic is complicated because it incorporates all of the sample data. While it is not easy to see the extension, the F statistic shown above is a generalization of the test statistic used for testing the equality of exactly two means.

This means that the outcome is equally variable in each of the comparison populations.


This assumption is the same as that assumed for appropriate use of the test statistic to test equality of two independent means. It is possible to assess the likelihood that the assumption of equal variances is true and the test can be conducted in most statistical computing packages. If the variability in the k comparison groups is not similar, then alternative techniques must be used.

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The F statistic is computed by taking the ratio of what is called the "between treatment" variability to the "residual or error" variability. This is where the name of the procedure originates. In analysis of variance we are testing for a difference in means H 0 : means are all equal versus H 1 : means are not all equal by evaluating variability in the data. The numerator captures between treatment variability i.

The test statistic is a measure that allows us to assess whether the differences among the sample means numerator are more than would be expected by chance if the null hypothesis is true. Recall in the two independent sample test, the test statistic was computed by taking the ratio of the difference in sample means numerator to the variability in the outcome estimated by Sp.

The decision rule again depends on the level of significance and the degrees of freedom. The F statistic has two degrees of freedom. These are denoted df 1 and df 2 , and called the numerator and denominator degrees of freedom, respectively. The degrees of freedom are defined as follows:. If the null hypothesis is true, the between treatment variation numerator will not exceed the residual or error variation denominator and the F statistic will small.

If the null hypothesis is false, then the F statistic will be large. The rejection region for the F test is always in the upper right-hand tail of the distribution as shown below. Because the computation of the test statistic is involved, the computations are often organized in an ANOVA table. The ANOVA table breaks down the components of variation in the data into variation between treatments and error or residual variation. The squared differences are weighted by the sample sizes per group n j. The error sums of squares is:.

The double summation SS indicates summation of the squared differences within each treatment and then summation of these totals across treatments to produce a single value. This will be illustrated in the following examples. The total sums of squares is:. If all of the data were pooled into a single sample, SST would reflect the numerator of the sample variance computed on the pooled or total sample. SST does not figure into the F statistic directly.

A clinical trial is run to compare weight loss programs and participants are randomly assigned to one of the comparison programs and are counseled on the details of the assigned program.

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Participants follow the assigned program for 8 weeks. The outcome of interest is weight loss, defined as the difference in weight measured at the start of the study baseline and weight measured at the end of the study 8 weeks , measured in pounds. Three popular weight loss programs are considered. The first is a low calorie diet. The second is a low fat diet and the third is a low carbohydrate diet.

For comparison purposes, a fourth group is considered as a control group. Participants in the fourth group are told that they are participating in a study of healthy behaviors with weight loss only one component of interest. The control group is included here to assess the placebo effect i. A total of twenty patients agree to participate in the study and are randomly assigned to one of the four diet groups. Weights are measured at baseline and patients are counseled on the proper implementation of the assigned diet with the exception of the control group.

After 8 weeks, each patient's weight is again measured and the difference in weights is computed by subtracting the 8 week weight from the baseline weight. Positive differences indicate weight losses and negative differences indicate weight gains. For interpretation purposes, we refer to the differences in weights as weight losses and the observed weight losses are shown below.

Is there a statistically significant difference in the mean weight loss among the four diets? The appropriate critical value can be found in a table of probabilities for the F distribution see "Other Resources". The critical value is 3. In order to compute the sums of squares we must first compute the sample means for each group and the overall mean based on the total sample.

SSE requires computing the squared differences between each observation and its group mean. We will compute SSE in parts. For the participants in the low calorie diet:. We reject H 0 because 8. ANOVA is a test that provides a global assessment of a statistical difference in more than two independent means. In this example, we find that there is a statistically significant difference in mean weight loss among the four diets considered. In addition to reporting the results of the statistical test of hypothesis i.

In this example, participants in the low calorie diet lost an average of 6. Participants in the control group lost an average of 1. Are the observed weight losses clinically meaningful? Who We Are. Hypothesis Testing: ANOVA Tests Hypothesis Testing uses statistics to choose between hypotheses regarding whether data is statistically significant or occurred by chance alone. One type of hypothesis tests are ANOVA tests, which are tests that examine whether two or more means are statistically significantly different from each other or whether the difference between them simply occurred by chance.

A One-Way ANOVA thus requires one categorical variable consisting of two or more groups, serving as the independent variable, and one continuous variable, serving as the dependent variable. The One-Way ANOVA is specifically looking at whether respondents of different subjective class identifications differ significantly in the mean number of college-level science classes taken. Then, select the nominal grouping variable of interest Subjective Class Identification, class from the list of variables and bring it over to the 'Factor' field.

One-way analysis of variance - Wikipedia

Then, click 'Continue. Here, we selected to include a Descriptives table and a Means plot in the output. Click 'Continue. The output consists of four parts.

The LSD test compares each group class category to all other groups class categories. Thus, please note that this table displays some comparisons more than once, since, in every row, each group is compared to all other groups. Each comparison is denoted by a differnet color, and lines of the same color represent repeated comparisons. In this example, the Upper Class differs significantly from the Lower, Working, and Middle Classes, and the Middle Class differs significantly from the Working Class in the number of college-level science courses taken.


The last section of the output, the Means Plot, is a graphical display of how the mean number of college-level science courses the respondents have taken depends on subjective class identification. A Factorial ANOVA thus requires one continuous variable to serve as the dependent variable and more than one categorical variable each consisting of two or more groups to serve as the independent variables.

In this example, the variable 'Subjective Class Identification, class' will be serving as the first categorical variable with 4 groups, and the variable 'Not Married, absingle' will be serving as the second categorical variable with 2 groups.