# One-way analysis of variance - ANOVA

Command: | Statistics ANOVA One-way analysis of variance |

## Description

One-way analysis of variance is used to test the difference between the means of several subgroups of a variable (multiple testing).

## How to enter data

The following figure illustrates how data need to be entered. For ANOVA, you need one continuous variable (concentration) and one qualitative variable (grade). The data for each case are entered on one row of the spreadsheet.

The qualitative factor may either be character or numeric codes. These codes are used to break-up the data into several subgroups for the ANOVA procedure, to calculate the *Between groups* and *Within groups* variation.

## Required input

For *Data*, select a continuous variable, and for *Factor codes* the qualitative factor.

### Options

**Logarithmic transformation**: if the data require a logarithmic transformation (e.g. when the data are positively skewed), select the Logarithmic transformation option.**Post-hoc test**: this is the test used for pairwise comparison of subgroups, when the ANOVA test is positive (i.e. P is less than the selected significance level, see below). MedCalc offers 3 post-hoc tests (in order of decreasing power): the Student-Newman-Keuls test, the Tukey-Kramer test and Scheffé's test.**Significance level**: the desired significance level for the post-hoc test. If the ANOVA test results in a P-value less than this significance level, MedCalc performs the selected post-hoc test.**Residuals**: you can select a Test for Normal distribution of the residuals.

## Results

### Levene's Test for Equality of Variances

Prior to the ANOVA test, Levene's Test for Equality of Variances is performed. If the Levene test is positive (P<0.05) then the variances in the different groups are different (the groups are not homogeneous) and you may need to apply a logarithmic transformation to the data, or use a non-parametric statistic.

### ANOVA

The results of the ANOVA are presented in an ANOVA table, followed by the F statistic and associated P value. If the P value is less than 0.05 (or another preselected significance level), then you can accept the hypothesis that the means of at least two of the subgroups differ significantly.

### Post-hoc test

If the ANOVA test is positive (P less than the selected significance level) then MedCalc performs a post hoc test (using either Student-Newman-Keuls', Tukey-Kramer's or Scheffé's method) for pairwise comparison of subgroups.

### Logarithmic transformation

If you selected the Logarithmic transformation option, the program performs the calculations on the logarithms of the dependent variable, but the different means are back-transformed and reported as the geometric means.

## Analysis of residuals

ANOVA analysis assumes that the residuals (the differences between the observations and the estimated values) follow a Normal distribution. This assumption can be evaluated with a formal test, or by means of graphical methods.

The different formal Tests for Normal distribution may not have enough power to detect deviation from the Normal distribution when sample size is small. On the other hand, when sample size is large, the requirement of a Normal distribution is less stringent because of the central limit theorem.

Therefore, it is often preferred to visually evaluate the symmetry and peakedness of the distribution of the residuals using the Histogram, Box-and-whisker plot, or Normal plot.

To do so, you click the hyperlink "Save residuals" in the results window. This will save the residual values as a new variable in the spreadsheet. You can then use this new variable in the different distribution plots.

### Graph

For a graphical representation of this test, refer to Multiple comparison graphs.

## Literature

- Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
- Armitage P, Berry G, Matthews JNS (2002) Statistical methods in medical research. 4
^{th}ed. Blackwell Science. - Sheskin DJ (2011) Handbook of parametric and non-parametric statistical procedures. 5
^{th}ed. Boca Raton: Chapman & Hall /CRC. - Snedecor GW, Cochran WG (1989) Statistical methods, 8
^{th}edition. Ames, Iowa: Iowa State University Press.