## Usage Note *35253: *Can I perform a one-way analysis of variance with only summary data in JMP®?

**Computing an Analysis of Variance with Summary Statistics in JMP® Software**

In some instances, an experimenter may want to perform an Analysis of Variance (ANOVA), but only the summary statistics are available. Most statistical programs are designed to compute ANOVA models with the full, complete set of data. However, David A. Larson describes a method to generate surrogate data from the summary statistics which can be used to fit the ANOVA of interest(1). That is, if the analysis is comparing k categories, and only the summary statistics (*n _{i}, mean_{i},s^{2}_{i} i= 1, 2, 3,..., k*) are available, then data can be generated to perform the desired analysis.

Utilizing Larson's ideas, JMP can perform this type of analysis. The following is an example demonstrating the appropriate steps taken to fit this ANOVA within JMP. If you have only two means to compare, JMP (beginning with version 11) provides an option in the Sample Data Index that can be used. Select **Help** ► **Sample Data**. In the Teaching resources section, open Calculators and click on Hypothesis Test for Two Means. Here, choose Summary Statistics and complete the dialog.

Now, for more than two means, suppose all the information available is the summary statistics below:

The first step is to create a JMP data table with the above data (Table 1).

**Table 1**: The JMP data table

According to Larson, two new columns need to be generated. So, create two columns named "Xi's" and "Xn's" having the **Formula** Column Property. Then, using JMP's Formula Editor, define these formulas:

Once these columns are created, they need to be "stacked." From the Tables menu, select Stack and select the columns "Xi's" and "Xn's" to be stacked, and also, change the name of the stacked column from the default "_Stacked_" to "Y" (Figure 1).

**Figure 1**: The "Stack" dialog box

The last item necessary to run the model is an appropriate frequency column added to the stacked data table. Using the If function (found in the Formula Editor's Conditional function list), create one more column named "Frequency" with the formula shown in Figure 2 below.

**Figure 2**: The "if" selection and the formula for "Frequency"

The final data table should appear as shown in Table 2.

**Table 2**: Final data table

The surrogate data has been generated so the ANOVA can now be performed. From the **Analyze** menu, choose **Fit Y by X**. Specify "Treatment" as "X", "Y" as "Y", and "Frequency" as "Freq," then click **OK** to run the analysis. The first item seen in the output is a scatterplot of the points. From the Oneway Analysis red triangle menu, click on Means/Anova to produce the resulting output seen in Figure 3.

**Figure 3**: ANOVA results

**Oneway ANOVA**

**Summary of Fit**

Rsquare | 0.690838 |

Adj Rsquare | 0.625752 |

Root Mean Square Error | 1.751986 |

Mean of Response | 15.85833 |

Observations (or Sum Wgts) | 24 |

**Analysis of Variance**

Source | DF | Sum of Squares | Mean Square | F Ratio |
Prob > F |

Treatment | 4 | 130.31833 | 32.5796 | 10.6141 | 0.0001 |

Error | 19 | 58.31965 | 3.0695 | ||

C. Total | 23 | 188.63798 |

**Means for Oneway ANOVA**

Level | Number | Mean |
Std Error | Lower 95% | Upper 95% |

A | 4 | 15.2000 | 0.8760 | 13.367 | 17.033 |

B | 6 | 12.8000 | 0.7152 | 11.303 | 14.297 |

C | 6 | 19.0000 | 0.7152 | 17.503 | 20.497 |

D | 5 | 17.1000 | 0.7835 | 15.460 | 18.740 |

E | 3 | 14.5000 | 1.0115 | 12.383 | 16.617 |

Std Error uses a pooled estimate of error variance.

As you see, the means are exactly those that were specified in the initial summary statistics. The standard errors given are estimated using a pooled estimate of the error variance. To compare all results, Table 3 gives the actual data from which the summary data is generated. The results from an analysis of variance using the actual data align perfectly to the output given with the summary statistics.

**Table 3**: Actual data

In conclusion, if only the summary statistics are available for an oneway analysis, the method described above can be followed to generate surrogate data in JMP to complete the desired analysis of variance.

**REFERENCES**

Larson, David A. (1992), "Analysis of Variance With Just Summary Statistics as Input," *American Statistician*, 46, 151-152.

#### Operating System and Release Information

Product Family | Product | System | SAS Release | |

Reported | Fixed* | |||

JMP Software | JMP software | Microsoft Windows 8.1 Enterprise 32-bit | ||

Microsoft Windows 8 Pro x64 | ||||

Microsoft Windows 8 Pro 32-bit | ||||

Microsoft Windows 8 Enterprise x64 | ||||

Microsoft Windows 8 Enterprise 32-bit | ||||

Microsoft® Windows® for x64 | ||||

Macintosh on x64 | ||||

Macintosh | ||||

Microsoft Windows 8.1 Enterprise x64 | ||||

Windows 7 Ultimate x64 | ||||

Windows 7 Professional x64 | ||||

Windows 7 Ultimate 32 bit | ||||

Windows 7 Home Premium 32 bit | ||||

Windows 7 Home Premium x64 | ||||

Windows 7 Professional 32 bit | ||||

Windows 7 Enterprise 32 bit | ||||

Windows 7 Enterprise x64 | ||||

Microsoft Windows 8.1 Pro 32-bit | ||||

Microsoft Windows 8.1 Pro x64 | ||||

Microsoft Windows 10 |

*****For software releases that are not yet generally available, the Fixed Release is the software release in which the problem is planned to be fixed.

Type: | Usage Note |

Priority: |

Date Modified: | 2016-04-22 12:56:00 |

Date Created: | 2009-03-23 14:39:20 |