I have got 2 models. The first model uses a factor as the independent variable. The 2nd model uses a matrix with 0/1 coefficients as the independent variable.
summary(lm(Petal.Width ~ 0+Species, data=iris))
smat = model.matrix(~ 0+Species, data=iris)
summary(lm(Petal.Width ~ 0+smat, data=iris))
The outcome of the two models above is the same. This is what I would expect:
Call:
lm(formula = Petal.Width ~ 0 + smat, data = iris)
Residuals:
Min 1Q Median 3Q Max
-0.626 -0.126 -0.026 0.154 0.474
Coefficients:
Estimate Std. Error t value Pr(>|t|)
smatSpeciessetosa 0.24600 0.02894 8.50 1.96e-14 ***
smatSpeciesversicolor 1.32600 0.02894 45.82 < 2e-16 ***
smatSpeciesvirginica 2.02600 0.02894 70.00 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2047 on 147 degrees of freedom
Multiple R-squared: 0.9796, Adjusted R-squared: 0.9792
F-statistic: 2357 on 3 and 147 DF, p-value: < 2.2e-16
However, when the model gets more complicated the summary is not the same anymore:
x = c(rep("x1",19),rep("x2",23),rep("x3",8),rep("x4",101),rep("x5",66),rep("x6",26),rep("x7",88))
y = c(0.74, 0.37, -0.34, 0.18, 0.18, -1.2, 0.04, 0.61, 0.85, 0, -0.03, -0.18, -0.12, 1.37, -0.34, -0.4, -1.71, -0.13, 3.63, 1.6, 0.47, 0.68, 0.14, 0.86, 0.68, 0.07, -0.33, 4.14, 5.3, 0.94, -0.54, -2.14, 0.12, 0.81, 1.51, -1.53, 3.37, -1.83, -6.99, -4.69, 1.59, -1.01, -0.09, 0.04, 0.01, 20.35, -0.28, -1.67, -1.08, -5.33, 1.26, 4.35, 0.07, 1.55, 0.65, 0.15, -0.08, -0.31, -1.61, 0.57, -0.36, 0.47, 0.72, -0.66, 0.92, 0.77, -1.93, -1.06, -0.93, 0.33, 0, 1.23, -1.53, -0.15, -3.5, 1.01, 0.63, 0.98, 1.06, 0.77, 0.33, 0.72, 1.56, 1.81, 0.74, 1.06, 0.09, 0.15, 0.75, 0.17, 0.77, 0.63, 0.54, 0.22, 0.49, -0.81, -0.03, 2.19, 0.76, 0.11, 0.94, -1.7, -2.11, 0.32, -4.9, 0.39, 1.59, -5.66, 2.08, -10.43, -0.08, -0.9, -0.16, -0.86, 0.43, -2.37, -2.16, -0.5, -0.6, 0.27, -0.34, 1.41, -0.42, -1.43, -2.25, -0.69, -0.71, -1.54, -3.12, -0.93, -2.01, 0.26, -0.53, -1.09, -1.45, -2.75, -1.63, 0.16, -0.13, 0.08, -2.93, -1.46, 0.18, -0.45, -0.88, 0.05, 0.68, -0.62, -2.55, -3.33, -2.35, -3.56, -0.07, -0.29, 1.81, 1.07, 0.01, 0.85, -0.21, -1.68, 1.13, 1.58, -0.31, 1.14, -0.73, 2.04, 0.82, 0.78, 0.69, 1.18, 3.78, 0.97, 0.7, 13.63, -2.35, 4.52, 7.46, 4.09, 1.36, 0.77, -0.51, 0.09, 3.12, -3.36, -1.04, 2.79, 0.1, 2.83, 1.96, 0.15, -0.09, 0.39, 0.65, -1.7, -0.84, 1.86, -0.6, 0.18, 0.49, -0.66, -0.16, 2.16, 0.46, 1.66, 0.13, 2.12, -0.03, -0.31, 0.13, -0.45, 0.18, -0.17, 0.25, -0.16, -0.06, 0.01, 0.83, -2.04, 1.06, 0.35, 0.86, 0.8, 2.01, 2.55, 0.07, 0.55, 1.59, -2.79, -2.77, 1.15, 0.46, 0.45, -0.4, 1.54, 0.29, 0.17, -0.36, 0.99, -9.59, 2.6, 1.79, -0.56, -0.49, 0.28, 1.46, -1.19, -0.48, -2.52, 1.08, -4.1, -2.91, -4.89, -0.12, -10.36, -4.97, -7.07, -3.81, -4.34, -12.82, -2.51, -8.47, -6.12, -1.56, -4.48, -2.88, -1.94, -6.01, -1.35, 1.23, 3.08, -2.3, -0.99, 0.63, 2.4, 2.27, -0.79, 1.76, 1.21, -4.43, -1.34, -1.94, -0.57, -2.77, -3.07, -1.15, -2.28, 0.68, 1.5, -1.78, 0.46, 0.39, -2.04, -1.22, 0.66, 1.22, 2.61, 5.21, 1.17, 2.73, 2.48, -0.95, 1.59, 0.14, -2.62, -1.73, -0.55, -1.25, 0.15, -0.11, -0.23, -3.47, 1.46, -1.32, 0.24, 1.48, 2.68, 0.07, 0.24, -0.76, -1.18, -1.57, -4.08, -2.74, 0.03, 0.34, 0.34, 0.84, 1.26, -0.81, -0.79, -1.59)
with the following output:
> summary(lm(y ~ 0+x))
Call:
lm(formula = y ~ 0 + x)
Residuals:
Min 1Q Median 3Q Max
-11.6615 -1.0202 0.0515 1.1498 18.8562
Coefficients:
Estimate Std. Error t value Pr(>|t|)
xx1 0.18526 0.59168 0.313 0.754
xx2 0.14000 0.53778 0.260 0.795
xx3 1.49375 0.91184 1.638 0.102
xx4 -0.41149 0.25663 -1.603 0.110
xx5 0.81182 0.31746 2.557 0.011 *
xx6 0.01077 0.50580 0.021 0.983
xx7 -1.15852 0.27493 -4.214 3.26e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.579 on 324 degrees of freedom
Multiple R-squared: 0.08401, Adjusted R-squared: 0.06422
F-statistic: 4.245 on 7 and 324 DF, p-value: 0.0001676
> summary(lm(y ~ 0+model.matrix(~x)))
Call:
lm(formula = y ~ 0 + model.matrix(~x))
Residuals:
Min 1Q Median 3Q Max
-11.6615 -1.0202 0.0515 1.1498 18.8562
Coefficients:
Estimate Std. Error t value Pr(>|t|)
model.matrix(~x)(Intercept) 0.18526 0.59168 0.313 0.7544
model.matrix(~x)xx2 -0.04526 0.79956 -0.057 0.9549
model.matrix(~x)xx3 1.30849 1.08699 1.204 0.2296
model.matrix(~x)xx4 -0.59675 0.64494 -0.925 0.3555
model.matrix(~x)xx5 0.62656 0.67147 0.933 0.3515
model.matrix(~x)xx6 -0.17449 0.77841 -0.224 0.8228
model.matrix(~x)xx7 -1.34379 0.65244 -2.060 0.0402 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.579 on 324 degrees of freedom
Multiple R-squared: 0.08401, Adjusted R-squared: 0.06422
F-statistic: 4.245 on 7 and 324 DF, p-value: 0.0001676
Why are the p-values of the coefficients different here?
If it's the order in which the coefficients are fitted, is there a way to get the same behaviour?
