In what way is model 1 equivalent to model 2 ? in what way is it different ? Bet

Question

In what way is model 1 equivalent to model 2 ? in what way is it different ? Between these two models, which one is better and why ?

Model1:

Call:
lm(formula = qualified ~ weight:relate, data = data)

Residuals:
   Min     1Q Median     3Q    Max
-3.65 -1.15 -0.15   0.85   2.85

Coefficients: (1 not defined because of singularities)
                Estimate Std. Error t value Pr(>|t|)   
(Intercept)       5.5926     0.1766 31.666 < 2e-16 ***
weight1:relate1 -0.9426     0.2707 -3.482 0.000632 ***
weight2:relate1 -0.4021     0.2670 -1.506 0.133908   
weight1:relate2 -0.4426     0.2707 -1.635 0.103934   
weight2:relate2       NA         NA      NA       NA   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.298 on 172 degrees of freedom
Multiple R-squared: 0.06599,   Adjusted R-squared: 0.0497
F-statistic: 4.051 on 3 and 172 DF, p-value: 0.008192

Model 2

Call:
lm(formula = qualified ~ weight + relate + weight:relate, data = data)

Residuals:
   Min     1Q Median     3Q    Max
-3.65 -1.15 -0.15   0.85   2.85

Coefficients:
                Estimate Std. Error t value Pr(>|t|)   
(Intercept)      4.65000    0.20521 22.660   <2e-16 ***
weight2          0.54048    0.28673   1.885   0.0611 .
relate2          0.50000    0.29021   1.723   0.0867 .
weight2:relate2 -0.09788    0.39435 -0.248   0.8043   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.298 on 172 degrees of freedom
Multiple R-squared: 0.06599,   Adjusted R-squared: 0.0497
F-statistic: 4.051 on 3 and 172 DF, p-value: 0.008192

Explanation / Answer

Both are equivalent in the sense, in both cases, the dependent

variable is qualified.

The first model takes only the interaction between weight and

relate into account i.e. it uses only the interaction as

independent variable to predict qualified.

The second model takes weight, relate as well as the

interactionbetween weight and relate into account as

independent variables.

In terms of co-efficients of determination, both the models

are same. But in the second model weight and relate are not

significant individually. So we should only consider ttheir

interaction. So we should choose Model 1. (Ans).

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