As described in One Sample Hypothesis Testing, the confidence interval is given by
estimated mean ± critical value ∙ std error
Alternatively by Property 1 of F Distribution, we can use the following as the critical value
We could have used as the estimate of the variance for the jth group in the calculation of the standard error, but since it is assumed that the variances of all the groups are equal, MSW is an estimate of group j based on a larger sample than , and so by the Law of Large Numbers, MSW provides a better estimate of the jth group variance than .
Example 1: Find the confidence intervals for each of the methods in Example 3 of Basic Concepts for ANOVA.
To do this we use the ANOVA data analysis tool found in the Real Statistics Resource Pack. Enter Ctrl-m and then double click on Analysis of Variance. Next select Anova: one factor from the dialog box that appears. as in Figure 0.
Figure 0 – Analysis of Variance dialog box
The dialog box shown in Figure 1 is now displayed.
Figure 1 – Dialog box for Single Factor Anova
Enter A3:D11 in the Input Range, select Excel format with column headings as the Input Format, select the ANOVA option and click on OK. The output appears as shown in Figure 2.
Figure 2 – Output of Real Statistics ANOVA data analysis tool
The confidence intervals are given in the range M7:N10. E.g. the confidence interval for Method 1 is calculated as follows:
tcrit = TINV(α, dfW) = TINV(.05, 25) = 2.06
Confidence interval = x̄i ± tcrit ∙ s.e. = 61.29 ± 2.06∙ 5.03 = (50.92, 71.65)
Note that most of the rest of the output in Figure 2 is similar to that found in the standard Excel data analysis tool (see, for example, Figure 5 of Basic Concepts for ANOVA). The entry for RMSSE is explained in Effect Size for ANOVA, while the entry for Omega Sq is explained in Other Measures of Effect Size for ANOVA.
Example 2: Conduct ANOVA for the data in the range A3:B31 of Figure 3 (only the first 20 elements are shown). This time the data is stored in what we will call stacked or standard format. Here the first column contains group names and the second column contains the corresponding scores. The data can be listed in any order.
Figure 3 – ANOVA for input in standard format
To conduct the analysis, click on cell D3 (where the output will start), enter Ctrl-m and select Analysis of Variance and press the OK button. Next select Single Factor Anova from the dialog box that appears. A dialog box will then appear similar to that shown in Figure 1. This time enter A3:B31 in the Input Range, select Standard format as the Input Format, deselect Columns/row headings included with data, select the ANOVA option and click on OK. The output appears in the rest of Figure 3.
Note that the data analysis tool first converts the input data from standard format to the usual Excel Anova format with column headings (range D5:G13). Since this data is the same as that used in Example 1, the ANOVA results are the same as shown in Figure 3.
Real Statistics Functions: The Real Statistics Resource Pack contains the following two array functions for converting between Single Factor Anova format and standard format.
StdAnova1(R1) = takes the data in R1 which is in standard format and outputs an array with the same data in Single Factor Anova format.
Anova1Std(R1): takes the data in R1 which is in Single Factor Anova format and outputs an array with the same data in standard format.
Observation: Referring to Figure 3, StdAnova1(A3:B31) yields the results shown in D5:G13. If you highlight the range S5:T35 and enter the array formula Anova1Std(D5:G3), the result is shown in Figure 4.
Figure 4 – Data in standard format
Note that the result is the same as range A3:B31, but in sorted order. Also note that any extra entries are filled in with #N/A.
Real Statistics Functions: In One-way ANOVA Basic Concepts we described the functions SSW, SSBet and SSTot, which can be used for data in Excel format. The following functions provide the same functionality for data in standard format.
SSWStd(R1, col) = SSW
SSBetStd(R1, col) = SSBet
SSTotStd(R1, col) = SSTot
Here the first column of range R1 contains the names of the factor levels and the other columns contain data values. The column number col contains the data for the one-way ANOVA that we want to perform. If col is omitted it defaults to 2 (the second column).
For the data in Figure 4, the formula =SSWStd(S5:T33) yields the value 4429.137, as seen in cell J17 of Figure 3.