In the article discussing 1-way analysis of variance (ANOVA), we compared the forces generated by 3 orthodontic wires. I used a sample of about 10 wires per group and measured the forces generated on a testing machine. Then I used 1-way ANOVA to assess whether there is a difference on average in the forces generated among any of the 3 wires.
Table I shows the ANOVA output and has been already interpreted in detail in the previous article.
| Source | SS | df | MS | F | P value |
|---|---|---|---|---|---|
| Between groups | 0.43 | 2 | 0.22 | 7.40 | 0.003 |
| Within groups | 0.76 | 26 | 0.03 | ||
| Total | 1.19 | 0.4 |
Briefly, the output indicates that there is a significant difference ( P = 0.003) in the forces generated between at least 2 wires.
Table II uses the same data set and variables to explore the same research question with a simple linear regression model instead of 1-way ANOVA. The findings are the same, but also the regression model includes estimates and confidence intervals that are always of interest.
| Wire type | β -coefficient | 95% CI | P value |
|---|---|---|---|
| Constant | 0.96 | 0.85, 1.07 | |
| Wire_2 | 0.29 | 0.12, 0.46 | 0.001 |
| Wire_3 | 0.22 | 0.07, 0.38 | 0.006 |
The interpretation of the regression model in Table II is similar to the previously discussed regression model. To clarify, please note that under the β-coefficient column, there are 2 values (wire_2 and wire_3) and the constant. The P values shown in the regression output are from the Wald test and correspond to the difference of each wire level to the baseline of the variable (wire_1). In Table I , the P value indicates the overall effect of wire type on the forces generated. The ANOVA P value indicates that the type of wire is a significant predictor of the forces generated; however, it does not tell us for which wires the forces generated differ significantly from each other. In the regression output, wire 1 is the reference category, and the coefficient values for wires 2 and 3 indicate the difference in forces generated between those wires and wire 1. Therefore, the force generated by wire 2 is 0.29 units higher than that of wire 1, and the force generated by wire 3 is 0.22 units higher than that of wire 1. The constant indicates the force generated by wire 1 (baseline or reference), and we can use this formula:
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