The references on this page includes the text book for the Javascript algorithm, and a web based teaching article on multiple regression. Other than these, it is assumed that users coming to this page will have other access to information and advice on the subject. The explanations and discussions that follows are therefore intended to help users follow the procedures and interpret the results, and not meant to be teaching or authoratative in nature.

Example used on this page

Mage	Mht	Gest	Sex	Bwt
Col 1	Col 2	Col 3	Col 4	Col 5
24	170	37	1	3048
29	161	36	0	2813
29	167	41	1	3622
21	165	36	1	2706
35	168	35	0	2581
27	161	39	0	3442
26	163	40	1	3453
34	167	37	0	3172
25	165	35	1	2386
28	170	39	0	3555
32	167	37	1	3029
31	169	37	0	3185
26	161	36	1	2670
21	165	38	0	3314
21	166	41	1	3596
24	164	38	0	3312
34	169	38	0	3414
25	161	41	0	3667
26	167	40	0	3643
27	162	33	1	1398
27	160	38	1	3135
21	167	39	1	3366

An example is provided to help users to follow the information and results in this page. It should be noted that the data presented here are artificially generated to demonstrate the procedures, and not real. Also that the sample size is deliberately small so they are easier to visualize, that in real life an adequate sample size will be very much larger. See subject index to find estimating sample size for multiple regression.

The example is to develop a model to predict birth weight of babies, using multiple regression. The independent predictive variables are maternal age in years (Mage), maternal height in cms (Mht), gestation age in weeks since beginning of pregnancy (Gest), sex of the baby (Sex, 0 for boys and 1 for girls). The dependent variable is birth weight in grams (Bwt). For this exercise we will use 22 cases. The data is presented in the table to the right.

Please note that, the Javascript program on this page allows any number of variables (columns of data), but designate the last column to the right as dependent variable. Again please note that many other variables are related to birth weight, and included in widely published models. Only 4 are included here to demonstrate the procedures and results

Correlation analysis

Correlation Coefficients Cols 1 2 3 4 5 1 1 0.2622 -0.2511 -0.3803 -0.0975 2 0.2622 1 0.0708 -0.1312 0.235 3 -0.2511 0.0708 1 -0.1089 0.9237 4 -0.3803 -0.1312 -0.1089 1 -0.3222 5 -0.0975 0.235 0.9237 -0.3222 1

Partial Correlation Coefficients Cols 1 2 3 4 5 1 -1 0.2185 -0.1468 -0.3183 0.0418 2 0.2185 -1 -0.3771 0.2392 0.4395 3 -0.1468 -0.3771 -1 0.4605 0.9493 4 -0.3183 0.2392 0.4605 -1 -0.5476 5 0.0418 0.4395 0.9493 -0.5476 -1

The program firstly calculates the correlation and partial correlation coefficients between all variables, and present them in tables as shown above. As well as serving any needs users may have, the purpose here is a check on correlations between the independent variables, as excessively close correlation between these may indicate that they may have same precursors, and risk introducing a bias.

The partial correlation coefficient reflects correlation between 2 variables after correcting for correlations with other variables in the data set. A large difference between partial and non-partial coefficients therefore reflect the possibility of excessive overlapping of measurements.

Multiple Regression

Col Reg (b) SEb t Sig (p,α) 1 1.70 9.86 0.17 0.8651 2 23.65 11.72 2.02 0.0598 3 223.19 17.92 12.45 <0.0001 4 -209.15 77.51 -2.70 0.0152 Constant(a) = -9165.48 Multiple Regression Coefficient R=0.961 F = 51.36, df=4,17, p<0.0001

Multiple regression analysis creates an equation which predicts birth weight (Bwt) from maternal age (Mage), maternal height (Mht), gestational age (Gest) and sex of the baby (Sex). The results are shown in the table to the right.

• Equation: y(col 5) = -9164.45(col 1) + 1.70(col 2) + 23.65(col 3) - 223.19(col 4)
• Col 1, Mage, increases Bwt by 1.7g per year, and is not statistically significant (p=0.97)
• Col 2, Mht, increases Bwt by 22.7g per cm, statistically significant at the p<0.1 level
• Col 3, Gest, increases Bwt by 223.2g per week, statistically highly significant at p<0.0001
• Col 4, Sex, decreases Bwt by 209.2g if its a girl, statistically significant at the p<0.05 level
• The equation, as a whole, has a multiple correlation coefficient of R=0.96, statistically highly significant with p<0.0001

Multiple Standardized Regression

Multiple standardized regression is the same as multiple regression, except that all measurements are standardized to the Standard Deviation unit z, where z = (value - mean) / SD. The coefficients produced are therefore of the same scale, making the structure of relationships between variables easier to visualize.

Each partial standardized regression coefficient (β) represents the change in the dependent variabe (y) in number of SDs, for each 1 SD change of the independent variable. The difference between the βs also reflect their relative influence on the dependent variable.

Col PSReg (β) SEβ t Sig (p,α) 1 0.01 0.08 0.17 0.8651 2 0.14 0.07 2.02 0.0598 3 0.90 0.07 12.45 <0.0001 4 -0.20 0.07 -2.70 0.0152 Constant(a) = 0 Multiple Regression Coefficient R=0.961 F = 51.36, df=4,17, p<0.0001

The results are as shown in the table to the right

• Equation: y(z₅) = 0.01(z₁) + 0.14(z₂) + 0.98(z₃) - 0.20(z₄)
• An incresse of 1 SD in maternal age affects 0.01 SD increase in birth weight, a trivial effect
• An incresse of 1 SD in maternal height affects 0.14 SD increase in birth weight, 14 times the effect of maternal age
• An incresse of 1 SD in gestational age affects 0.98 SD increase in birth weight, almost in tandem, 98 times the effect of maternal age and 7 time that of maternal height
• An incresse of 1 SD in Sex affects 0.2 SD decrease in birth weight, 20 times the effect of maternal age, 1.4 times that of maternal height, and 0.2 times (20%) that of gestation

References

Steel RGD, Torrie JH, Dickey DA (1997) Principles and procedures of statistics. A biomedical approach. 3rd Ed. McGraw-Hill Inc New York NY 10020 ISBN 0-07-061028-2 p. 322-351

https://wiki.gis.com/wiki/index.php/Multiple_Regression a detailed explanation of multiple regression available on line

Data Entry
The data is a matrix of measurements
  - Each row from a case
  - Each column a variable
  - The last column to the right is the dependent variable y in multiple regression

The following is a single program, divided into parts so it is easier to follow

Part 1. Data entry

# Data entry to dataframe
myDat = ("
Mage     Mht   Gest     Sex     Bwt
24	 170	37	 1	3048
29       161	36	 0	2813
29	 167	41	 1	3622
21	 165	36	 1	2706
35	 168	35	 0	2581
27	 161	39	 0	3442
26	 163	40	 1	3453
34	 167	37	 0	3172
25	 165	35	 1	2386
28	 170	39	 0	3555
32	 167	37	 1	3029
31	 169	37	 0	3185
26	 161	36	 1	2670
21	 165	38	 0	3314
21	 166	41	 1	3596
24	 164	38	 0	3312
34	 169	38	 0	3414
25	 161	41	 0	3667
26	 167	40	 0	3643
27	 162	33	 1	1398
27	 160	38	 1	3135
21	 167	39	 1	3366         ") 
df <- read.table(textConnection(myDat),header=TRUE)      
#summary(df)  # optional display of input data

Part 2: Means and Standard Deviations

# mean and SD
meanMage = mean(df$Mage)
sdMage = sd(df$Mage)
meanMht = mean(df$Mht)
sdMht = sd(df$Mht)
meanGest = mean(df$Gest)
sdGest = sd(df$Gest)
meanSex = mean(df$Sex)
sdSex = sd(df$Sex)
meanBwt = mean(df$Bwt)
sdBwt = sd(df$Bwt)

# show means and Sds
c(meanMage,sdMage)     
c(meanMht,sdMht)     
c(meanGest,sdGest)     
c(meanSex,sdSex)     
c(meanBwt,sdBwt)   

The means and SD results are as follows

> # show means and Sds
> c(meanMage,sdMage)     
[1] 26.954545  4.281491
> c(meanMht,sdMht)     
[1] 165.227273   3.191235
> c(meanGest,sdGest)     
[1] 37.772727  2.136571
> c(meanSex,sdSex)     
[1] 0.5000000 0.5117663
> c(meanBwt,sdBwt)     
[1] 3113.955  532.697

Part 3: Multiple regression

RegRes<-lm(Bwt~Mage+Mht+Gest+Sex,data=df)    # Multiple regression
summary(RegRes)     # show multiple regression results 

The results are as follos

> summary(RegRes)     # show multiple regression results 

Call:
lm(formula = Bwt ~ Mage + Mht + Gest + Sex, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-469.89  -86.47   46.49   84.17  198.44 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -9165.476   1945.069  -4.712 0.000201 ***
Mage            1.701      9.864   0.172 0.865124    
Mht            23.649     11.724   2.017 0.059759 .  
Gest          223.194     17.920  12.455 5.68e-10 ***
Sex          -209.150     77.511  -2.698 0.015228 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 163.7 on 17 degrees of freedom
Multiple R-squared:  0.9236,	Adjusted R-squared:  0.9056 
F-statistic: 51.36 on 4 and 17 DF,  p-value: 2.849e-09

Part 4: Repeat Multiple regression using Standardized values

Standardized values z = (value-mean) / SD

Part 4a: Create standardized z values

# standardization
# create z variables
df$ZMage <- (df$Mage - meanMage) / sdMage
df$ZMht <- (df$Mht - meanMht) / sdMht
df$ZGest <- (df$Gest - meanGest) / sdGest
df$ZSex <- (df$Sex - meanSex) / sdSex
df$ZBwt <- (df$Bwt - meanBwt) / sdBwt

Part 4b: Standardized multiple regression using z values

RegZRes<-lm(ZBwt~ZMage+ZMht+ZGest+ZSex,data=df)    # Multiple regression
summary(RegZRes)     # show multiple regression results 

The results are as follows. For all variables mean = 0 and SD = 1

> summary(RegZRes)     # show multiple regression results

Call:
lm(formula = ZBwt ~ ZMage + ZMht + ZGest + ZSex, data = df)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.88209 -0.16233  0.08728  0.15800  0.37252 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.267e-16  6.551e-02   0.000   1.0000    
ZMage        1.367e-02  7.928e-02   0.172   0.8651    
ZMht         1.417e-01  7.024e-02   2.017   0.0598 .  
ZGest        8.952e-01  7.188e-02  12.455 5.68e-10 ***
ZSex        -2.009e-01  7.447e-02  -2.698   0.0152 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3073 on 17 degrees of freedom
Multiple R-squared:  0.9236,	Adjusted R-squared:  0.9056 
F-statistic: 51.36 on 4 and 17 DF,  p-value: 2.849e-09

To make the coefficients easier to read,it is trans;ated as follows

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)          0     0.0655  0.0000   1.0000    
ZMage           0.0137     0.0793  0.1720   0.8651    
ZMht            0.1417     0.0702  2.0170   0.0598 .  
ZGest           0.8952     0.0719 12.4550  <0.0001 ***
ZSex           -0.2009     0.0745 -2.6980   0.0152 *