Fitting a multiple linear regression in R

Question

So I have data like this -

##      V2   V3     V4   V5   V6   V7    V8
## 2  27.0 41.3 2948.0 26.2 51.7 42.7  89.8
## 3  22.9 66.7 4644.0  3.0 45.7 41.8 121.3
## 4  26.3 58.1 3665.0  3.0 50.8 38.5 115.2
## 5  29.1 39.9 2878.0 18.3 51.5 38.8 100.3
## 6  28.1 62.6 4493.0  7.0 50.8 39.7 123.0
## 7  26.2 63.9 3855.0  3.0 50.7 31.1 124.8

I want to do a multiple linear regression -

model1 = lm(cigarette.data$V8 ~ cigarette.data$V2 + cigarette.data$V3 + cigarette.data$V4 + cigarette.data$V5 + cigarette.data$V6 + cigarette.data$V7, data = cigarette.data)

But this gives me -

    ## 
## Call:
## lm(formula = cigarette.data$V8 ~ cigarette.data$V2 + cigarette.data$V3 + 
##     cigarette.data$V4 + cigarette.data$V5 + cigarette.data$V6 + 
##     cigarette.data$V7, data = cigarette.data)
## 
## Residuals:
## ALL 51 residuals are 0: no residual degrees of freedom!
## 
## Coefficients: (186 not defined because of singularities)
##                         Estimate Std. Error t value Pr(>|t|)
## (Intercept)                   19         NA      NA       NA
## cigarette.data$V223.1         20         NA      NA       NA
## cigarette.data$V223.9         23         NA      NA       NA
## cigarette.data$V224.8        -16         NA      NA       NA
## cigarette.data$V225.0         21         NA      NA       NA
## cigarette.data$V225.1         25         NA      NA       NA
## cigarette.data$V225.9         -9         NA      NA       NA
## cigarette.data$V226.2          8         NA      NA       NA

Which seems wrong. What's going on?

Answer 1

The problem is that you are fitting a model with more predictor variables than samples (ie rows). Your example contains 6 samples, so 5 variables (+ intercept = 6) would predict the V8 predictand perfectly:

cigarette.data <- structure(list(V2 = c(27, 22.9, 26.3, 29.1, 28.1, 26.2), V3 = c(41.3, 
66.7, 58.1, 39.9, 62.6, 63.9), V4 = c(2948, 4644, 3665, 2878, 
4493, 3855), V5 = c(26.2, 3, 3, 18.3, 7, 3), V6 = c(51.7, 45.7, 
50.8, 51.5, 50.8, 50.7), V7 = c(42.7, 41.8, 38.5, 38.8, 39.7, 
31.1), V8 = c(89.784450178314, 121.359442280557, 115.031032135658, 
100.201279353697, 123.401631728502, 124.750887806)), .Names = c("V2", 
"V3", "V4", "V5", "V6", "V7", "V8"), row.names = c(NA, -6L), class = "data.frame")

fit <- lm(V8 ~ V2 + V3 + V4 + V5 + V6 + V7, data = cigarette.data)
summary(fit)


Call:
lm(formula = V8 ~ V2 + V3 + V4 + V5 + V6 + V7, data = cigarette.data)

Residuals:
ALL 6 residuals are 0: no residual degrees of freedom!

Coefficients: (1 not defined because of singularities)
            Estimate Std. Error t value Pr(>|t|)
(Intercept) 98.89203         NA      NA       NA
V2           5.66196         NA      NA       NA
V3           2.16574         NA      NA       NA
V4          -0.01412         NA      NA       NA
V5           0.03093         NA      NA       NA
V6          -4.07376         NA      NA       NA
V7                NA         NA      NA       NA

Residual standard error: NaN on 0 degrees of freedom
Multiple R-squared:      1, Adjusted R-squared:    NaN 
F-statistic:   NaN on 5 and 0 DF,  p-value: NA

Your model should either contain fewer variables or more samples (see example below):

fit <- lm(V8 ~ V2 + V3 + V4 + V5, data = cigarette.data)
summary(fit)

Call:
lm(formula = V8 ~ V2 + V3 + V4 + V5, data = cigarette.data)

Residuals:
      1       2       3       4       5       6 
-1.1873  0.9570 -2.9738  1.9870 -0.7142  1.9312 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept) 17.846025  57.297709   0.311    0.808
V2           1.848628   1.240164   1.491    0.376
V3           0.802375   0.879204   0.913    0.529
V4           0.001821   0.008315   0.219    0.863
V5          -0.583697   0.601185  -0.971    0.509

Residual standard error: 4.4 on 1 degrees of freedom
Multiple R-squared:  0.981, Adjusted R-squared:  0.9052 
F-statistic: 12.94 on 4 and 1 DF,  p-value: 0.2052

Answer 2

There must be a null value or 0.0 as one of the records in your data frame. Try to impute those records or remove it from your data frame before fitting the model.