[R] Multiple regression in R - unstandardised coefficients a
JC Matthews
J.C.Matthews at bristol.ac.uk
Tue Aug 23 13:54:28 CEST 2011
Thankyou for your replies, you've answered my question and given me more to
think on. I guess it is unwise to draw any conclusions from the
standardised results for these reasons.
James.
--On 22 August 2011 17:30 +0100 ted.harding at wlandres.net wrote:
> On 22-Aug-11 15:37:40, JC Matthews wrote:
>> Hello,
>>
>> I have a statistical problem that I am using R for, but I am
>> not making sense of the results. I am trying to use multiple
>> regression to explore which variables (weather conditions)
>> have the greater effect on a local atmospheric variable.
>> The data is taken from a database that has 20391 data points (Z1).
>>
>> A simplified version of the data I'm looking at is given below,
>> but I have a problem in that there is a disagreement in sign
>> between the regression coefficients and the standardised regression
>> coefficients. Intuitively I would expect both to be the same sign,
>> but in many of the parameters, they are not.
>>
>> I am aware that there is a strong opinion that using standardised
>> correlation coefficients is highly discouraged by some people,
>> but I would nevertheless like to see the results. Not least
>> because it has made me doubt the non-standardised values of B
>> that R has given me.
>>
>> The code I have used, and some of the data, is as follows (once
>> the database has been imported from SQL, and outliers removed).
>>
>> Z1sub <- Z1[, c(2, 5, 7,11, 12, 13, 15, 16)]
>> colnames(Z1sub) <- c("temp", "hum", "wind", "press", "rain", "s.rad",
>> "mean1", "sd1" )
>>
>> attach(Z1sub)
>> names(Z1sub)
>>
>>
>> Model1d <- lm(mean1 ~ hum*wind*rain + I(hum^2) + I(wind^2) + I(rain^2)
>> )
>>
>> summary(Model1d)
>>
>> Call:
>> lm(formula = mean1 ~ hum * wind * rain + I(hum^2) + I(wind^2) +
>> I(rain^2))
>>
>> Residuals:
>> Min 1Q Median 3Q Max
>> -1230.64 -63.17 18.51 97.85 1275.73
>>
>> Coefficients:
>> Estimate Std. Error t value Pr(>|t|)
>> (Intercept) -9.243e+02 5.689e+01 -16.246 < 2e-16 ***
>> hum 2.835e+01 1.468e+00 19.312 < 2e-16 ***
>> wind 1.236e+02 4.832e+00 25.587 < 2e-16 ***
>> rain -3.144e+03 7.635e+02 -4.118 3.84e-05 ***
>> I(hum^2) -1.953e-01 9.393e-03 -20.793 < 2e-16 ***
>> I(wind^2) 6.914e-01 2.174e-01 3.181 0.00147 **
>> I(rain^2) 2.730e+02 3.265e+01 8.362 < 2e-16 ***
>> hum:wind -1.782e+00 5.448e-02 -32.706 < 2e-16 ***
>> hum:rain 2.798e+01 8.410e+00 3.327 0.00088 ***
>> wind:rain 6.018e+02 2.146e+02 2.805 0.00504 **
>> hum:wind:rain -6.606e+00 2.401e+00 -2.751 0.00594 **
>> ---
>> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1
>> ' ' 1
>>
>> Residual standard error: 180.5 on 20337 degrees of freedom
>> Multiple R-squared: 0.2394, Adjusted R-squared: 0.239
>> F-statistic: 640.2 on 10 and 20337 DF, p-value: < 2.2e-16
>>
>>
>>
>>
>>
>> To calculate the standardised coefficients, I used the following:
>>
>> Z1sub.scaled <- data.frame(scale( Z1sub[,c('temp', 'hum', 'wind',
>> 'press',
>> 'rain', 's.rad', 'mean1', 'sd1' ) ] ) )
>>
>> attach(Z1sub.scaled)
>> names(Z1sub.scaled)
>>
>>
>> Model1d.sc <- lm(mean1 ~ hum*wind*rain + I(hum^2) + I(wind^2) +
>> I(rain^2) )
>>
>> summary(Model1d.scaled)
>>
>> Call:
>> lm(formula = mean1 ~ hum * wind * rain + I(hum^2) + I(wind^2) +
>> I(rain^2))
>>
>> Residuals:
>> Min 1Q Median 3Q Max
>> -5.94713 -0.30527 0.08946 0.47287 6.16503
>>
>> Coefficients:
>> Estimate Std. Error t value Pr(>|t|)
>> (Intercept) 0.0806858 0.0096614 8.351 < 2e-16 ***
>> hum -0.4581509 0.0073456 -62.371 < 2e-16 ***
>> wind -0.1995316 0.0073767 -27.049 < 2e-16 ***
>> rain -0.1806894 0.0158037 -11.433 < 2e-16 ***
>> I(hum^2) -0.1120435 0.0053885 -20.793 < 2e-16 ***
>> I(wind^2) 0.0172870 0.0054346 3.181 0.00147 **
>> I(rain^2) 0.0040575 0.0004853 8.362 < 2e-16 ***
>> hum:wind -0.2188729 0.0066659 -32.835 < 2e-16 ***
>> hum:rain 0.0267420 0.0146201 1.829 0.06740 .
>> wind:rain 0.0365615 0.0122335 2.989 0.00281 **
>> hum:wind:rain -0.0438790 0.0159479 -2.751 0.00594 **
>> ---
>> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1
>> ' ' 1
>>
>> Residual standard error: 0.8723 on 20337 degrees of freedom
>> Multiple R-squared: 0.2394, Adjusted R-squared: 0.239
>> F-statistic: 640.2 on 10 and 20337 DF, p-value: < 2.2e-16
>>
>>
>>
>> So having, for instance for humidity (hum), B = 28.35 +/- 1.468, while
>> Beta = -0.4581509 +/- 0.0073456 is concerning. Is this normal, or is
>> there
>> an error in my code that has caused this contradiction?
>>
>> Many thanks,
>>
>> James.
>> ----------------------
>> JC Matthews
>> School of Chemistry
>> Bristol University
>
> Hi,
> without having your data, so unable to check, I would not be
> surprised if the changes of sign were the outcome of your model
> formula, in particular the 3-variable (2nd-order) interaction,
> i.e. you are using a model which is non-linear in the variables
> themselves. Let's just take that part of the model:
>
> lm(formula = mean1 ~ hum * wind * rain
>
> This, in its quantitative expression, expands to:
>
> mean1 = C0 + C11*hum + C12*wind + C13*rain
> + C21*hum*wind + C22*hum*rain + C23*wind*rain
> + C31*hum*wind*rain
>
> Suppose that is for the unstandardised variables. Now express
> it in terms of standardised variables (initial capital letters):
>
> mean1 = C0 + C11*sd(hum)*(Hum + mean(hum)/sd(hum))
> + C12*sd(wind)*(Wind + mean(wind)/sd(wind))
> + C13*sd(rain)*(Rain + mean(rain)/sd(rain))
>
> + C21*sd(hum)*sd(wind)*
> (Hum + mean(hum)/sd(hum))*(Wind + mean(wind)/sd(wind))
>
> + C22*sd(hum)*sd(rain)*
> (Hum + mean(hum)/sd(hum))*(Rain + mean(rain)/sd(rain))
>
> + C23*sd(wind)*sd(rain)*
> (Wind + mean(wind)/sd(wind))*
> (Rain + mean(rain)/sd(rain))
>
> + C31*sd(hum)*sd(wind)*sd(rain)*
> (Hum + mean(hum)/sd(hum))*
> (Wind + mean(wind)/sd(wind))*
> (Rain + mean(rain)/sd(rain))
>
> Now pick out, say, the coefficient of 'Hum' in this latter expression
> (i.e. all the terms which involve 'Hum' but neither 'Wind' nor 'Rain'):
>
> C11*sd(hum)
> + C21*sd(hum)*sd(wind)*mean(wind)/sd(wind)
> + C22*sd(hum)*sd(rain)*mean(rain)/sd(rain)
> + C31*sd(hum)*sd(wind)*sd(rain)*
> (mean(wind)/sd(wind))*(mean(rain)/sd(rain))
>
> = C11*sd(hum)
> + C21*sd(hum)*mean(wind)
> + C22*sd(hum)*mean(rain)
> + C31*sd(hum)*mean(wind)*mean(rain)
>
> So there is no reason to expect this to have even the same sign
> as the original C11, the coefficient of 'hum', let alone any more
> specific relationship with it!
>
> Hoping this helps,
> Ted.
>
>
>
> --------------------------------------------------------------------
> E-Mail: (Ted Harding) <ted.harding at wlandres.net>
> Fax-to-email: +44 (0)870 094 0861
> Date: 22-Aug-11 Time: 17:30:29
> ------------------------------ XFMail ------------------------------
----------------------
JC Matthews
Atmospheric Chemistry Research Group
School of Chemistry
Bristol University
J.C.Matthews at bristol.ac.uk
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