## Abstract

*Journal*provide real dietetics-related illustrations of how the techniques have been used in practice. Figure 1 provides definitions of terms used throughout this article.

## When to Use ANOVA

### The *t* test

*t*test is used to determine whether there is a statistically significant difference in LDL levels between the experimental group and control group. What if a third group of 20 subjects was assigned to a Mediterranean diet and their LDL was 100 mg/dL (2.6 mmol/L)? What statistical test can be used to determine whether there is a statistically significant difference between three groups rather than two? The

*t*test is limited to comparing the means of two groups. At its most basic, ANOVA is used to compare the means of two or more groups for statistically significant differences.

### The Basics of ANOVA

### The *F* Ratio

*F*ratio rather than the

*t*value that is used to compare the means of two groups using the

*t*test. The

*F*ratio is the explained variance divided by the unexplained in a data set. The explained variance is also called the between-group variance and the unexplained variance is called the within-group variance. Figure 2 presents a dataset that illustrates the concepts of within- and between-group variance. As presented in Figure 2, the between-group variance is that which exists between group means and within-group variance is that which exists within each separate group in a data set. If there is no difference between group means then the between-group and within-group variances are equal, yielding

*F*=1. If there are statistically significant differences between groups, then the between-group variance is greater than within, yielding an

*F*>1. This appears to be the case in Figure 2. The variation in group means appears to be significantly greater than the variation of values within the groups.

*F*ratio is calculated it is compared to a critical value based on a set level of significance, usually 0.05 (a 5% probability that an observed difference between group means is by chance, and not a real difference). If the

*F*ratio exceeds the critical value then there is a statistically significant difference between the means of the groups. These concepts will be operationalized with an example in the section on one-way ANOVA. The

*F*ratio is the test statistic for all types of ANOVA.

## One-Way ANOVA

*t*test for independent samples. The

*t*test for independent samples tests if there is a statistically significant difference between means of two independent samples for a specific quantitative variable. When the means of more than two samples are to be compared one-way ANOVA is applied.

*F*ratio. A high

*F*ratio indicates the greater likelihood of a statistically significant difference between groups.

### The One-Way ANOVA Calculation

*F*ratio can be determined so true statistical differences can be determined. Table 1 presents the calculations of the

*F*ratio. In Table 1, the between-group mean square is the between-group variance and the within-groups mean square is the within-group variance. When the former is divided by the latter it yields

*F*=69—a value much greater than 1—indicating that the between-group variance is substantially higher than the within-group variance. Therefore, the variation in treatments explains a significant amount of variance in LDL levels. The

*F*ratios that must be exceeded for levels of significance of 0.05 and 0.01 for the statistically significant differences in LDL level based on treatment are 2.87 and 4.38, respectively. These two values are selected from an

*F*ratio table based on the between- and within-group degrees of freedom (Table 1) (

*F*ratios that must be exceeded for confidence that there is either <5% or 1% probability that the differences in means between the groups are by chance, rather than real differences. Sixty-nine significantly exceeds these two values indicating that the mean LDL levels of the four treatment groups are significantly different.

Source of variation | Sum of squares | df | Mean square | F | P value |
---|---|---|---|---|---|

Between-group variance | 11,248 | 3 | 3,749 | 69 | <0.001 |

Within-group variance | 1,971 | 36 | 55 | ||

Total | 13,219 | 39 |

### Interpreting the Results

*F*ratio, the researcher would not know for sure if mean LDL values for the group taking the LDL-lowering drug differ significantly from the group following the Mediterranean diet. Follow-up tests must be done to determine the groups that are actually different from one another. These tests are called post hoc tests with names such as least square difference, Scheffé test, Tukey-Kramer test, Duncan multiple range test, Fisher exact test, Newman-Keuls test, and Dunnett test. In all variations of ANOVA, post hoc tests must be done to ferret out the specific categories in which there are actual differences. These post hoc tests allow multiple pairwise comparisons of means among the groups being compared. The mean LDL values for the four groups mentioned above are 87 mg/dL (2.26 mmol/L) (LDL-lowering drug), 107 mg/dL (2.78 mmol/L) (Mediterranean diet), 133 mg/dL (3.46 mmol/L) (American diet), and 100 mg/dL (2.6 mmol/L) (vegan diet). A post hoc test allows the determination of which of these means are statistically different from one another. When the Tukey-Kramer test is applied to this situation it reveals statistically significant differences between all the means but the Mediterranean and vegan diets. The LDL-lowering drug is superior to all treatments in producing a lower LDL level with the Mediterranean and vegan diets intermediate, and the American diet least effective. With one-way ANOVA it is important to know not only that the groups are statistically different but also which groups are different from one another. Investigators would want to know which approach led to the lowest LDL levels. For an example of the application of one-way ANOVA refer to the study by Smith and colleagues (

## Repeated-Measures ANOVA

### Application to Repeated Measurements Taken Over Time

*t*test (also called

*t*test for dependent samples).

*F*ratio is the test statistic.

*F*ratio indicates statistical significance a post hoc test must be done to determine at which points in the time sequence of LDL levels there are statistically significant differences. Suppose the mean values are 130 mg/dL (3.38 mmo/L) pre-program, and 120 mg/dL (3.12 mmol/L), 110 mg/dL (2.86 mmol/L), 100 mg/dL (2.6 mmol/L), 90 mg/dL (2.34 mmol/L), 91 mg/dL (2.36 mmol/L), 95 mg/dL (2.63 mmol/L), 95 mg/dL (2.63 mmol/L), and 92 mg/dL (2.39 mmol/L) at the consecutive 1-month follow-up periods. As in the case of the one-way ANOVA, conducting a post hoc test such as the Tukey-Kramer test will enable the investigator to determine which of the means are significantly different. For example, is there a statistically significant difference between means of 130 mg/dL (3.38 mmol/L) and 120 mg/dL (3.12 mmol/L)? How about between the means of 90 mg/dL (2.34 mmo/L) and 91 mg/dL (2.36 mmol/L)? The post hoc tests indicate which means are statistically different. Maybe the most significant changes occurred during the 4-month program, but stabilized thereafter?

### Application to Subjects Receiving Multiple Treatments

*F*ratio is statistically significant then a post hoc test would be done to determine which burger had the best rating.

Subject | Nut-based burger | Soy-based burger | Grain-based burger |
---|---|---|---|

1 | 7 | 5 | 3 |

2 | 8 | 6 | 2 |

3 | 9 | 6 | 3 |

## Multiway ANOVA

*F*ratio that enables the investigator to determine the magnitude of each effect.

Condition | Placebo | Vitamin D-3 supplementation |
---|---|---|

←ng/mL→ | ||

No sunlight | −5 | 15 |

Sun exposure | 20 | 65 |

*F*ratio. As noted previously, the magnitude of the

*F*ratio is an indicator of the statistical significance of the interaction.

*F*ratios calculated for the main effects and the interactions (see Table 4 for an example of an ANOVA statistical table). In the vitamin D-3 example, it appears from the means (Table 3) that there is a disproportionate effect of the combined treatments compared to any alone; hence, an interaction (Figure 3) . So the answer to, “What is the effect of vitamin D-3 supplementation,” the answer is it depends on whether the person gets sunlight or not. In other words, the effect of the sunlight is dependent on whether or not subject received supplements. In this case there is a synergistic effect of sunlight and supplementation, not just main effects. If there was no interaction, supplementation and sunlight exposure could have independent effects, but no joint effects.

Source of Variation | Sum of squares | df | Mean square | F | P value |
---|---|---|---|---|---|

Supplement | 5,281.25 | 1 | 5,281.25 | 281.67 | <0.0005 |

Sunlight | 7,031.25 | 1 | 7,031.25 | 375.00 | <0.0005 |

Supplement × sunlight | 781.25 | 1 | 781.25 | 41.67 | <0.0005 |

Error | 300.00 | 16 | 18.75 | ||

Total | 13,393.75 | 19 |

*F*ratios for the main effects and interactions. If the

*F*ratio for the interaction is significant then it is irrelevant to report any main effects, because the variables are not independent and so cannot have independent main effects.

## ANCOVA

### Adjusting Post-Test Measurements for Pre-Test Differences

*F*ratio will be generated as the test statistic and can be evaluated for statistical significance. ANCOVA can adjust outcome variables for differences between groups on quantitative variables. If this

*F*value is statistically significant, then ANCOVA will have told us that there is a statistically significant difference in posttest blood pressure between the experimental and control groups after adjusting for the potentially confounding effect of the pre-test blood pressure.

Group | Pretest diastolic heart rate | Post-test diastolic heart rate |
---|---|---|

←mm Hg→ | ||

Hypertension control program | 90 | 70 |

Control | 110 | 110 |

*Super Size Me*on fast-food knowledge, psychosocial measures, and awareness-raising effectiveness scores among young adults.

### Adjusting the Relationship between an Independent and Dependent Variable for Another Confounding Independent Variable

## MANOVA

### Reasons for Using MANOVA

^{5}=0.23. By running five separate one-way ANOVAs, there is a 23% chance that the tests will yield differences by chance rather than be real differences. This is too great of a chance. In statistics it is accepted practice to avoid running multiple tests on similar concepts with the same sample or samples. Running these relationships all at once reduces the likelihood that differences will arise by chance.

### Application of MANOVA

*F*ratio if the user is more comfortable with this more common test statistic.

*F*ratios for the individual ANOVAs.

## MANCOVA

## Conclusions

## References

- Table of critical values for the F distribution (for use with ANOVA).(University of Sussex Web site) (Accessed June 21, 2011)
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