One-Way and Two-Way ANOVA in ExcelLarry A. Pace, Ph.D.

Let us learn to conduct and interpret a one-way ANOVA and a two-way ANOVA for a balanced factorial design in Excel, using the Analysis ToolPak. We will perform the calculations by use of formulas and compare the results with those from the ToolPak and from SPSS. As a bonus, the reader will have access to a worksheet template for the two-way ANOVA that automates all the calculations required for Module 6, Assignment 2.One-way ANOVAThe analysis of variance (ANOVA) allows us to compare three or more means simultaneously, while controlling for the overall probability of making a Type I error (rejecting a true null hypothesis). Researchers developed a test to measure participants pain thresholds. The researchers sought to determine whether participants hair color had any influence on her pain tolerance. Twenty participants were divided into four equal-sized groups based on natural, and their pain tolerance was measured. The dependent variable is the pain tolerance score, and the independent variable (or factor) is the participants hair color. In ANOVA, the independent variable is the grouping variable that determines group membership for each participant. Consider the following hypothetical pain threshold data. Higher scores indicate higher pain tolerance.Hair Color

Light BlondeDark BlondeLight BrownDark Brown

62634232

60575039

71524151

55413730

48434335

The null hypothesis is that the population means are equal. The alternative hypothesis is that at least one pair of means is unequal. We can symbolize these hypotheses as follows:H0: 1 = 2 = 3 = 4H1: 1 2 3 4Partitioning the Total Sum of SquaresAnalysis of variance partitions or analyzes the total variation into between-groups variation (based on differences between the means of the groups) and within-groups variation (based on differences between the individual values and the mean of the group to which that value belongs). Symbolically:SStot = SSb + SSwThe total sum of squares, SStot is found (by definition) as follows. Find the overall mean of all observations, ignoring the group membership. Let us use x double bar to represent the mean of all the observations, which is sometimes called the grand mean.

We have 4 groups with 5 observations each, so we have 20 total observations. Treating all 20 observations as a single dataset, the mean is 47.6. To find the total sum of squares, subtract the mean from every value, square this deviation score, and then add up all the squared deviations:

and thus,

Although this is instructive, it is quite laborious and is not the way we would compute the sum of squares by hand. Instead, we would use a computing form that is algebraically equivalent to our definitional form, but requires us simply to square and sum our raw score values, rather than to find individual deviation scores.

The sum of squares total can be found by summing all the squared values, and subtracting from that total the sum of the individual values squared and divided by N, the total number of observations. This formula gives us exactly the same value as the definitional formula. See the Excel screen shot below. The selected cell, C16, contains the formula for squaring the sum of the x values and dividing the square by N. When we subtract that value from the sum of the squares of all the scores, the remainder is the total sum of squares.

Once you understand what the total sum of squares represents, you can easily use technology to find it without having to do repetitive calculations. Use the DEVSQ function in Excel to find the sum of squares from raw data. See the formula in the Formula Bar and the resulting value in cell E7. To get this answer, simply click in cell E7 and type the formula exactly as you see it in the Formula Bar.

Now that we have the total sum of squares, let us partition it into two different sources. The between-groups sum of squares is based only on the differences between the group means and the overall mean. We weight the group mean by the number of observations in the group (because that is the number of observations that went into calculating the mean) and multiply the squared deviation from the overall mean for that group by the group size. Add this up across all groups, and you have the between-groups sum of squares. It is easier to show this by use of a formula than to write it in words. Let us call the number of groups k. In our case, k = 4. Further, let us say there are n1 + n1++nk = N total observations. In our case, we have 5 + 5 + 5 + 5 = 20 observations. The definitional formula of the between-groups sum of squares is:

For our pain threshold data, we would have the following results.

The within-groups sum of squares is found by summing the squared deviations from the mean for each score in that group, and then summing across groups. The double summation just says find the squared deviations for each group and then add them all up across the groups.

The table below shows the calculation of the within-groups sum of squares.

Because of the additivity principle of variance, we could just as easily have found the within-groups sum of squares by simple subtraction:SStot = SSb + SSwSStot - SSb = SSw2384.8 1382.8 = 1002Partitioning the Degrees of FreedomYou will recall the concept of degrees of freedom (number of values free to vary) from your module on t tests. In ANOVA, we partition the total degrees of freedom in a fashion similar to the way we partition the total sum of squares. Remember the basic n minus one definition for degrees of freedom. If you have 20 total observations, you have 19 total degrees of freedom. If you have 4 groups, you have 3 degrees of freedom between groups. If you have 5 observations in a group, you have 4 degrees of freedom in that group. Using the same symbols we have already discussed, we have N 1 total degrees of freedom. We have k 1 degrees of freedom between groups, and we have N k degrees of freedom within groups. The table below shows the partition of the total sum of squares and of the degrees of freedom.

Mean Squares and the F RatioDividing a sum of squares by its degrees of freedom produces a mean square or MS. A mean square is a variance estimate, and we calculate an F ratio by dividing two variance estimates. The MSb is the variance due to differences between the means of the groups, and the MSw is the variance due to differences within the groups. By calculating an F ratio, we determine how large the between-group variance is, relative to the within-group variance. Let us build this additional information into our ANOVA summary table.

If the two variance estimates were equal, the F ratio would be 1. As the F ratio increases, the amount of variation due to real differences between the groups increases. As with a t test, we can find the probability of obtaining an F ratio as large as or larger than the one we obtained if the null hypothesis of no differences is true. We will compare the p value we obtain to the alpha level for our test, which we usually set to .05. If our p value is lower than .05, we reject the null hypothesis. If the p value is greater than .05, we retain the null hypothesis.Using our current example, let us complete our ANOVA summary table.

To test the significance of an F ratio of 7.36 with 3 and 16 degrees of freedom, we can use Excels FDIST function. The function takes three arguments, which are the value of F, the degrees of freedom for the numerator term, and the degrees of freedom for the denominator term. We see that our F ratio has a p value much lower than .05, so our decision is to reject the null hypothesis.

Writing the Results of the ANOVA in APA FormatIn our APA summary statement, we do not say we rejected the null hypothesis, but instead that the results are statistically significant, which is another way of saying the same thing. An APA-style conclusion for our analysis might be as follows:A one-way ANOVA revealed that there are significant differences in pain thresholds among women of different hair color, F(3, 16) = 7.36, p = .003.Using the Analysis ToolPak for a One-Way ANOVAFirst, ensure you have the Analysis ToolPak installed. If you do, there will be a Data Analysis option in the Analysis group of the Data ribbon.

If you do not see this option, click on the Office Button, then Excel Options > Add-ins > Manage Excel Add-ins > Go. In the resulting dialog box, check the box in front of Analysis ToolPak, and click OK.

To conduct the one-way ANOVA, first make sure your data are in a worksheet arranged as follows. The labels and borders are purely optional. Because we will use labels, we must inform Excel of that fact.

Click on Data > Analysis > Data Analysis. In the Data Analysis dialog box, scroll to Anova: Single Factor, and then click OK.

Enter the input range by dragging through the entire dataset, including the labels. Check Labels in First Row, and then click OK. The results of the ANOVA will appear in a new worksheet.

I have formatted the tables number formatting to be consistent with APA style.

Note the Analysis ToolPak produces the same values as we did with our manual calculations. An ANOVA summary table from SPSS is shown below. SPSS produces the same results as the Analysis ToolPak. Note SPSS uses the label Sig. for significance as a label for the p value.

Going Further: Effect SizeBecause our ANOVA is significant, we might ask the very reasonable question, How big is the effect? One very good way to answer this question is to calculate an effect size index known as 2 (eta squared). This index tells us what proportion of the total variation in the dependent variable can be explained (or accounted for) by knowing the independent variable. In our case, we would ask what proportion of variation in pain tolerance can be explained by knowing a womans hair color. We calculate 2 by dividing the between-groups sum of squares by the total sum of squares. In our example, 1382.8 / 2384.8 = .58, so a substantial amount of the variation is explained.Going Further: Post Hoc ComparisonsAfter a significant ANOVA, we know that at least one pair of means is different, but we do not yet know which pair or pairs are significantly different. We want to control the probability of making a Type I error, so we want to use a post hoc comparison procedure that holds the overall or experimentwise error rate to no more than our original alpha level. Two very popular post hoc procedures are the Tukey HSD (for honestly significant difference) procedure and Bonferroni-corrected comparisons. The Tukey HSD test uses a distribution called the studentized range statistic, while the Bonferroni procedure uses the t distribution with which you are already familiar. For that reason, we will illustrate the Bonferroni procedure. We will calculate a new value of t using the following formula:

We find the difference between two of the means, and divide that by a standard error term based on all the groups rather than just the two groups in consideration. We weight the error term by the sizes of the two groups, but for the t test, we use the within-groups degrees of freedom instead of n1 + n2 2 as we would in an independent-samples t test. Then, to make the Bonferroni correction, we divide the alpha level for the overall ANOVA by the number of possible pairings of means. Because we have 4 groups, there are 6 possible pairings:

Thus we would have / 6 = .05 / 6 = .0083 as the required level of significance to reject the null hypothesis that the two means in question are different. If we wanted to do this in reverse using the p value approach, we could find the p value for our t test with dfw, and then multiply that p value by the number of possible pairings. Let us work out one example. We will compare the means of light blondes and dark brunettes:

Using Excels TDIST function, we find this value of t has a two-tailed p value of .0005 with 16 degrees of freedom.

This is lower than .0083, and we can reject the null hypothesis and conclude that light blondes have a significantly higher pain tolerance threshold than dark brunettes. If we use the p value approach, we can multiply .0005 by 6 to find the actual p value to be approximately .003. Rather than doing all these corrected t tests by hand, we might use a technology such as SPSS to perform all the pairwise comparisons. Note in the SPSS output that the mean difference, the standard error term, and the p value for the comparison of light blondes and dark brunettes all agree with our calculations above.

The Two-Way ANOVAIn the one-way ANOVA, we have a single factor. In our example, the factor is the grouping variable hair color. In the two-way ANOVA, we have two factors. For example, we might have one factor that as before is hair color and another factor that is sex. In our first example, we had females only. What if we repeated our experiment, but added males? In the two-way ANOVA, we can determine whether there are sex differences, whether there are differences by hair color, and whether there is an interaction between sex and hair color. We will consider only the simplest of cases, a balanced factorial design with equal numbers of observations in every cell of the data table. Assume we have the following data:

Partitioning the Total Sum of Squares in Two-Way ANOVAIn the two-way ANOVA, we can define and calculate the total sum of squares in an identical fashion to the one-way ANOVA. We now have a sum of squares for A, the row factor, a sum of squares for B, the column factor, a sum of squares for the interaction between the A and B factors, and a residual (for what is left over) sum of squares that we will use as our error term. Our partition isSStot = SSA + SSB + SSAB + SSerrSimply calculate the overall or grand mean, subtract that mean from every individual observation, square the deviation score, and sum the squared deviations.

Let us change our symbols slightly to accommodate the present case. Let C represent the number of columns (our B factor), and R (for rows) represent the number of levels of our A factor. Let G (for group) represent the number of observations per cell. Then we have N = C R G. In our example there are 2 levels of A, 4 levels of B, and 5 observations per cell, so N = 2 4 5 = 40. Thus we have 40 1 = 39 total degrees of freedom.We calculate the sum of squares for cells as the sum of the squared deviations of the individual cell means from the overall mean multiplied by the number of observations per group. This is the overall between groups variation, which we partition further into SSA, SSB and SSAB.

Use the marginal means for the row factor A, ignoring the column factor B, to calculate the sum of squares for A:

with R 1 degrees of freedom. In a similar fashion, the sum of squares for factor B, ignoring the row factor A, is

with G 1 degrees of freedom. Subtracting the sum of squares for A and B from the sum of squares for cells gives the sum of squares for the interaction of A and B,

with (R 1) (C 1) degrees of freedom. Finally, we can also find the error sum of squares by subtraction:

with RC (G 1) degrees of freedom. We calculate the mean square for each term by dividing the sum of squares by the respective degrees of freedom and then calculate the F ratio for each effect by dividing the mean square for that effect by the error mean square. Because these calculations are laborious, we will usually allow technology to do them for us, though the learner may benefit from working a small example by hand. Let us do the calculations for our revised example. We can use the DEVSQ function in Excel to find the total degrees of freedom from the raw data, as we did in our one-way ANOVA.

Below is a table of the cell and marginal means for our dataset. Let us use our A and B labeling to make sure we understand the row and column factors.

We can use this information to help us calculate the various sums of squares. The sum of squares for cells is the number of observations per cell multiplied by the deviations for each cell from the overall mean of all observations. There are 5 observations per cell, so the cell sum of squares is

The cell sum of squares is an intermediate step for finding the AB interaction sum of squares, so we will now calculate the sum of squares for A, our row factor. We will multiply the number of columns by the number of observations per cell, and multiply this by the squared deviations of the row marginal means from the overall mean.

Similarly, we find the column sum of squares by multiplying the number of rows by the number of observations per cell, and multiply that by the squared deviations of the column marginal means from the overall mean.

We can now find the AB interaction sum of squares by subtraction:

We can also find the error sum of squares by subtraction:

In the two-way ANOVA we will have the following ANOVA summary table with the degrees of freedom as shown. Note there are three F ratios, one for the A main effect, one for the B main effect, and one for the AB interaction. Each F ratio uses the same error term, which is the error mean square.

Using the information from our current example, we have the following summary table:

Note there is a significant main effect for sex (Factor A), and a significant main effect for hair color (Factor B). The interaction of hair color and sex is not significant. We might write our results as follows:A two-way ANOVA revealed a significant main effect for sex, F(1, 32) = 142.47, p < .001, and a significant main effect for hair color, F(3, 32) = 13.01, p < .001. There was no interaction between sex and hair color, F(3, 32) = 0.44, p = .723.Doing the Two-Way ANOVA in the Analysis ToolPak Make sure the Analysis ToolPak is installed. Input the data as shown below. I will use A and B to reinforce the concept of row and column factors as discussed above.

Select Data > Analysis > Data Analysis. In the Data Analysis dialog box, scroll to Anova: Two-factor With Replication.

Click OK, and in the resulting dialog, drag through the entire range of data, including the labels. The rows per sample indicate the number of observations per cell.

Click OK. Excel conducts the ANOVA procedure and places the results in a new worksheet (shown after cell formatting). The ANOVA summary table appears below. Excel calls the row factor Samples, and the column factor Columns. Apart from this labeling difference, the results are identical to those we calculated with the definitional formulas.

Going Further: Two-Way ANOVA in SPSS with Bonferroni ComparisonsThe two-way ANOVA in SPSS produces exactly the same results as the Analysis ToolPak.

Because there are only two levels of A, there is no need for additional comparisons for that factor. But there are four levels of B (hair color), so we can compare the means for the levels of B using Bonferroni comparisons. We see again that regardless of sex, people with light blond hair have higher pain tolerance than people with light and dark brunette hair. Further, we see that people with dark blonde hair have higher pain tolerance than those with dark brunette hair.

Bonus: A Two-Way ANOVA Worksheet TemplateI have created a two-way ANOVA template in Excel. This template requires only that the user type or paste the data in the appropriate cells. The template is programmed to have as many as 10 observations per cell, and will work with up to five levels of a row factor A and 10 levels of a column factor B. The current data correctly entered into the template are as follows:

As another bonus, the template calculates partial 2 (partial eta squared) as an appropriate effect-size index for each effect in the two-way ANOVA. Note the summary table from the template agrees with our hand calculations, the Analysis ToolPak, and SPSS.

To use the template, simply type or paste the data into the green-shaded data entry area for each level of the row factor A and the column factor B, starting with observation 1 for each level of A. The ANOVA summary table appears to the right of the data entry area.