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I performed both a linear and log-linear regression to predict the price of a smartphone based on its characteristics. Now I have a question concerning the coefficients between the two models.

In the linear regression model, the dummy variable GPS included or not is 37,7. This means that smartphone users pay on average 47.7 euro more for a smartphone with a GPS built in than one without, while holding other variables in the model constant.

lm <- lm(Price ~ ., data=data_price2)
summary(lm)

Call:
lm(formula = Price ~ ., data = data_price2)

Residuals:
Min      1Q  Median      3Q     Max 
-702.43  -46.68   -6.49   37.59 1522.53 

Coefficients: (38 not defined because of singularities)
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)              44.62802   70.21355   0.636 0.525128    
Screensize               -6.78973    7.14553  -0.950 0.342155    
Multitouch               11.20542   12.62356   0.888 0.374861    
nbrCores                 14.58104    2.67044   5.460 5.53e-08 ***
Processorspeed           46.84652    9.54521   4.908 1.02e-06 ***
Memory                  -24.12829    6.02706  -4.003 6.54e-05 ***
nbrSims                  -9.23095    8.00187  -1.154 0.248842    
CameraBack                3.10923    0.62724   4.957 7.94e-07 ***
CameraFront              10.69124    2.45340   4.358 1.40e-05 ***
Autofocus               -20.51415    9.40548  -2.181 0.029326 *  
Flitsertype              10.63140    7.10996   1.495 0.135043    
5-GHzOndersteuning             NA         NA      NA       NA    
GPS                      47.68043   11.81778   4.035 5.73e-05 ***
....
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 102.3 on 1556 degrees of freedom
Multiple R-squared:  0.7766,    Adjusted R-squared:  0.7613 
F-statistic: 51.02 on 106 and 1556 DF,  p-value: < 2.2e-16

Next, when we take a look at the log-linear regression model, the coefficient for the GPS variable is 2.249e-02, which means that the smartphone retail price increases with 2.52% = (e2.249e-02 − 1) when GPS is included, while holding other variables in the model constant.

lm3 <- lm(log(Price) ~ ., data = data_price2 )
summary(lm3)

Call:
lm(formula = log(Price) ~ ., data = data_price2)

Residuals:
Min      1Q  Median      3Q     Max 
-2.3367 -0.1964 -0.0008  0.1896  3.1645 

Coefficients: (38 not defined because of singularities)
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)             3.268e+00  2.598e-01  12.575  < 2e-16 ***
Screensize              4.878e-02  2.644e-02   1.845 0.065255 .  
Multitouch              2.155e-02  4.672e-02   0.461 0.644685    
nbrCores                5.670e-02  9.883e-03   5.737 1.16e-08 ***
Processorspeed          7.306e-02  3.533e-02   2.068 0.038787 *  
Memory                  8.273e-03  2.231e-02   0.371 0.710761    
nbrSims                -3.488e-02  2.961e-02  -1.178 0.239022    
CameraBack              9.779e-03  2.321e-03   4.213 2.67e-05 ***
CameraFront             5.348e-02  9.080e-03   5.890 4.73e-09 ***
Autofocus               1.061e-02  3.481e-02   0.305 0.760654    
Flitsertype             1.080e-01  2.631e-02   4.105 4.26e-05 ***
5-GHzOndersupport              NA         NA      NA       NA    
GPS                     2.249e-02  4.374e-02   0.514 0.607221    
....
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3785 on 1556 degrees of freedom
Multiple R-squared:  0.7974,    Adjusted R-squared:  0.7835 
F-statistic: 57.76 on 106 and 1556 DF,  p-value: < 2.2e-16

The average price for a smartphone in my model is 232€. So, in the log-linear model 2.52% of 232€ is +- 5.85€. How come this value is so different in comparison with the result obtained from the linear regression model?

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  • $\begingroup$ Please clarify what you mean by "log-linear regression," as that term can have a technical meaning that is not the same as a linear regression with a log-transformed outcome variable. It would help a lot to see more details of the data (maybe some plots), the summaries of the regression outputs, and so forth. $\endgroup$ – EdM Jul 3 '16 at 16:24
  • $\begingroup$ I've added the regression outputs from both the linear regression and the regression with the log-transformed outcome variable $\endgroup$ – GerritCalle Jul 7 '16 at 9:33
  • $\begingroup$ Why do you have so many singularities? $\endgroup$ – mdewey Jul 7 '16 at 16:03
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It's not just GPS, whose coefficient is "significant" in the linear price model but not in the log-price model. Many of your predictors change in apparent "significance" between the two models: screen size, memory, auto focus, flitsertype, too.

This is probably due to significant correlations among sets of your predictors, called multicollinearity. In that situation, exactly which predictor gets "credit" for an influence on outcome (in terms of a significant regression coefficient) depends strongly on peculiarities of the data sample at hand. Best guess is that the log transformation of price simply changed which variables among the multicollinear set happened to get that "credit." The distressingly large number of coefficients "not defined because of singularities" might even represent perfect correlations among some predictors.

Depending on what you're trying to accomplish, you might be better off using ridge regression, which better handles multicollinearity as it tends to treat correlated predictors together. But first look closely at the relations among your predictors, remove any that are perfectly correlated to other predictors, and think hard about how you want to deal with sets of highly correlated predictors.

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