## Problem

You want to perform linear regressions and/or correlations.

## Solution

Some sample data to work with:

# Make some data
# X increases (noisily)
# Z increases slowly
# Y is constructed so it is inversely related to xvar and positively related to xvar*zvar
set.seed(955)
xvar <- 1:20 + rnorm(20,sd=3)
zvar <- 1:20/4 + rnorm(20,sd=2)
yvar <- -2*xvar + xvar*zvar/5 + 3 + rnorm(20,sd=4)

# Make a data frame with the variables
dat <- data.frame(x=xvar, y=yvar, z=zvar)
# Show first few rows
#>           x           y           z
#> 1 -4.252354   4.5857688  1.89877152
#> 2  1.702318  -4.9027824 -0.82937359
#> 3  4.323054  -4.3076433 -1.31283495
#> 4  1.780628   0.2050367 -0.28479448
#> 5 11.537348 -29.7670502 -1.27303976
#> 6  6.672130 -10.1458220 -0.09459239

### Correlation

# Correlation coefficient
cor(dat\$x, dat\$y)
#> [1] -0.7695378

### Correlation matrices (for multiple variables)

It is also possible to run correlations between many pairs of variables, using a matrix or data frame.

# A correlation matrix of the variables
cor(dat)
#>            x            y           z
#> x  1.0000000 -0.769537849 0.491698938
#> y -0.7695378  1.000000000 0.004172295
#> z  0.4916989  0.004172295 1.000000000

# Print with only two decimal places
round(cor(dat), 2)
#>       x     y    z
#> x  1.00 -0.77 0.49
#> y -0.77  1.00 0.00
#> z  0.49  0.00 1.00

To visualize a correlation matrix, see ../../Graphs/Correlation matrix.

### Linear regression

Linear regressions, where dat\$x is the predictor, and dat\$y is the outcome. This can be done using two columns from a data frame, or with numeric vectors directly.

# These two commands will have the same outcome:
fit <- lm(y ~ x, data=dat)  # Using the columns x and y from the data frame
fit <- lm(dat\$y ~ dat\$x)     # Using the vectors dat\$x and dat\$y
fit
#>
#> Call:
#> lm(formula = dat\$y ~ dat\$x)
#>
#> Coefficients:
#> (Intercept)        dat\$x
#>     -0.2278      -1.1829

# This means that the predicted y = -0.2278 - 1.1829*x

# Get more detailed information:
summary(fit)
#>
#> Call:
#> lm(formula = dat\$y ~ dat\$x)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -15.8922  -2.5114   0.2866   4.4646   9.3285
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  -0.2278     2.6323  -0.087    0.932
#> dat\$x        -1.1829     0.2314  -5.113 7.28e-05 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 6.506 on 18 degrees of freedom
#> Multiple R-squared:  0.5922,	Adjusted R-squared:  0.5695
#> F-statistic: 26.14 on 1 and 18 DF,  p-value: 7.282e-05

To visualize the data with regression lines, see ../../Graphs/Scatterplots (ggplot2) and ../../Graphs/Scatterplot.

### Linear regression with multiple predictors

Linear regression with y as the outcome, and x and z as predictors.

Note that the formula specified below does not test for interactions between x and z.

# These have the same result
fit2 <- lm(y ~ x + z, data=dat)    # Using the columns x, y, and z from the data frame
fit2 <- lm(dat\$y ~ dat\$x + dat\$z)  # Using the vectors x, y, z
fit2
#>
#> Call:
#> lm(formula = dat\$y ~ dat\$x + dat\$z)
#>
#> Coefficients:
#> (Intercept)        dat\$x        dat\$z
#>      -1.382       -1.564        1.858

summary(fit2)
#>
#> Call:
#> lm(formula = dat\$y ~ dat\$x + dat\$z)
#>
#> Residuals:
#>    Min     1Q Median     3Q    Max
#> -7.974 -3.187 -1.205  3.847  7.524
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  -1.3816     1.9878  -0.695  0.49644
#> dat\$x        -1.5642     0.1984  -7.883 4.46e-07 ***
#> dat\$z         1.8578     0.4753   3.908  0.00113 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.859 on 17 degrees of freedom
#> Multiple R-squared:  0.7852,	Adjusted R-squared:  0.7599
#> F-statistic: 31.07 on 2 and 17 DF,  p-value: 2.1e-06

#### Interactions

The topic of how to properly do multiple regression and test for interactions can be quite complex and is not covered here. Here we just fit a model with x, z, and the interaction between the two.

To model interactions between x and z, a x:z term must be added. Alternatively, the formula x*z expands to x+z+x:z.

# These are equivalent; the x*z expands to x + z + x:z
fit3 <- lm(y ~ x * z, data=dat)
fit3 <- lm(y ~ x + z + x:z, data=dat)
fit3
#>
#> Call:
#> lm(formula = y ~ x + z + x:z, data = dat)
#>
#> Coefficients:
#> (Intercept)            x            z          x:z
#>      2.2820      -2.1311      -0.1068       0.2081

summary(fit3)
#>
#> Call:
#> lm(formula = y ~ x + z + x:z, data = dat)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -5.3045 -3.5998  0.3926  2.1376  8.3957
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  2.28204    2.20064   1.037   0.3152
#> x           -2.13110    0.27406  -7.776    8e-07 ***
#> z           -0.10682    0.84820  -0.126   0.9013
#> x:z          0.20814    0.07874   2.643   0.0177 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.178 on 16 degrees of freedom
#> Multiple R-squared:  0.8505,	Adjusted R-squared:  0.8225
#> F-statistic: 30.34 on 3 and 16 DF,  p-value: 7.759e-07