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JC Matthews

2011-Aug-22 15:37 UTC

[R] Multiple regression in R - unstandardised coefficients are a different sign to standardised coefficients, is this correct?

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

Ista Zahn

2011-Aug-22 16:02 UTC

head link

[R] Multiple regression in R - unstandardised coefficients are a different sign to standardised coefficients, is this correct?

Hi JC,

You have interactions in your model, which means that your models
specifies that the coefficients for hum, wind, and rain should vary
depending on the value of the other two (and depending on their own
value actually, since you also have quadratic effects for each of
these variables in your model). Since these coefficients are varying
according to the model, it is impossible to specify their value
unconditionally. The values you are seeing are therefore conditional
estimates that at particular values on the variables with which each
predictor interacts. Since you've changed the distribution of those
variables by standardizing them, you get different conditional
estimates.

All this will be covered in most regression textbooks.

Best,
Ista

On Mon, Aug 22, 2011 at 11:37 AM, JC Matthews
<J.C.Matthews at bristol.ac.uk> 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
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>


-- 
Ista Zahn
Graduate student
University of Rochester
Department of Clinical and Social Psychology
http://yourpsyche.org

(Ted Harding)

2011-Aug-22 16:30 UTC

head link

[R] Multiple regression in R - unstandardised coefficients a

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 ------------------------------

Ista Zahn

2011-Aug-23 13:54 UTC

head link

[R] Multiple regression in R - unstandardised coefficients a

On Tue, Aug 23, 2011 at 7:54 AM, JC Matthews <J.C.Matthews at
bristol.ac.uk> wrote:> 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.
No, by all means try to draw conclusions! Isn't that the point of the
analysis in the first place? All I am (we are?) saying is that you
need to do your homework and learn how to draw _appropriate_
conclusions from the analysis.

Best,
Ista
>
> 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
>


-- 
Ista Zahn
Graduate student
University of Rochester
Department of Clinical and Social Psychology
http://yourpsyche.org

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