# Decoding the Mystery: How to Interpret Regression Output in R Like a Champ

rtip
regression
Author

Steven P. Sanderson II, MPH

Published

December 14, 2023

# Introduction

Ever run an R regression and stared at the output, feeling like you’re deciphering an ancient scroll? Fear not, fellow data enthusiasts! Today, we’ll crack the code and turn those statistics into meaningful insights.

Let’s grab our trusty R arsenal and set up the scene:

• Dataset: `mtcars` (a classic car dataset in R)
• Regression: Linear model with `mpg` as the dependent variable (miles per gallon) and all other variables as independent variables (predictors)

# Step 1: Summon the Stats Gods with “summary()”

First, cast your R spell with `summary(lm(mpg ~ ., data = mtcars))`. This incantation conjures a table of coefficients, p-values, and other stats. Don’t panic if it looks like a cryptic riddle! We’ll break it down:

``````model <- lm(mpg ~ ., data = mtcars)

summary(model)``````
``````
Call:
lm(formula = mpg ~ ., data = mtcars)

Residuals:
Min      1Q  Median      3Q     Max
-3.4506 -1.6044 -0.1196  1.2193  4.6271

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 12.30337   18.71788   0.657   0.5181
cyl         -0.11144    1.04502  -0.107   0.9161
disp         0.01334    0.01786   0.747   0.4635
hp          -0.02148    0.02177  -0.987   0.3350
drat         0.78711    1.63537   0.481   0.6353
wt          -3.71530    1.89441  -1.961   0.0633 .
qsec         0.82104    0.73084   1.123   0.2739
vs           0.31776    2.10451   0.151   0.8814
am           2.52023    2.05665   1.225   0.2340
gear         0.65541    1.49326   0.439   0.6652
carb        -0.19942    0.82875  -0.241   0.8122
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.65 on 21 degrees of freedom
Multiple R-squared:  0.869, Adjusted R-squared:  0.8066
F-statistic: 13.93 on 10 and 21 DF,  p-value: 3.793e-07``````

## Coefficients

These tell you how much, on average, the dependent variable changes for a one-unit increase in the corresponding independent variable (holding other variables constant). For example, a coefficient of 0.05 for `cyl` means for every one more cylinder, mpg is expected to increase by 0.05 miles per gallon, on average.

``model\$coefficients``
``````(Intercept)         cyl        disp          hp        drat          wt
12.30337416 -0.11144048  0.01333524 -0.02148212  0.78711097 -3.71530393
qsec          vs          am        gear        carb
0.82104075  0.31776281  2.52022689  0.65541302 -0.19941925 ``````

## P-values

These whisper secrets about significance. A p-value less than 0.05 would mean the observed relationship between the variable and mpg is unlikely to be due to chance. The following are the individual p-values for each variable:

``summary(model)\$coefficients[, 4]``
``````(Intercept)         cyl        disp          hp        drat          wt
0.51812440  0.91608738  0.46348865  0.33495531  0.63527790  0.06325215
qsec          vs          am        gear        carb
0.27394127  0.88142347  0.23398971  0.66520643  0.81217871 ``````

Now the overall p-value for the model:

``````model_p <- function(.model) {

# Get p-values
fstat <- summary(.model)\$fstatistic
p <- pf(fstat[1], fstat[2], fstat[3], lower.tail = FALSE)
print(p)
}

model_p(.model = model)``````
``````       value
3.793152e-07 ``````

# Step 2: Let’s Talk Turkey - Interpreting the Numbers

## Coefficients

Think of them as slopes. A positive coefficient means the dependent variable increases with the independent variable. Negative? The opposite! For example, `disp` has a negative coefficient, so bigger engines (larger displacement) tend to have lower mpg.

## P-values

Imagine a courtroom. A low p-value is like a strong witness, convincing you the relationship between the variables is real. High p-values (like for `am`!) are like unreliable witnesses, leaving us unsure.

# Step 3: Zoom Out - The Bigger Picture

## R-squared

This tells you how well the model explains the variation in mpg. A value close to 1 is fantastic, while closer to 0 means the model needs work. In our case, it’s not bad, but there’s room for improvement.

``summary(model)\$r.squared``
``[1] 0.8690158``

## Residuals

These are the differences between the actual mpg values and the model’s predictions. Analyzing them can reveal hidden patterns and model issues.

``data.frame(model\$residuals)``
``````                    model.residuals
Mazda RX4              -1.599505761
Mazda RX4 Wag          -1.111886079
Datsun 710             -3.450644085
Hornet 4 Drive          0.162595453
Valiant                -2.283039036
Duster 360             -0.086256253
Merc 240D               1.903988115
Merc 230               -1.619089898
Merc 280                0.500970058
Merc 280C              -1.391654392
Merc 450SE              2.227837890
Merc 450SL              1.700426404
Merc 450SLC            -0.542224699
Lincoln Continental    -0.536437711
Chrysler Imperial       4.206370638
Fiat 128                4.627094192
Honda Civic             0.503261089
Toyota Corolla          4.387630904
Toyota Corona          -2.143103442
Dodge Challenger       -1.443053221
AMC Javelin            -2.532181498
Camaro Z28             -0.006021976
Pontiac Firebird        2.508321011
Fiat X1-9              -0.993468693
Porsche 914-2          -0.152953961
Lotus Europa            2.763727417
Ford Pantera L         -3.070040803
Ferrari Dino            0.006171846
Maserati Bora           1.058881618
Volvo 142E             -2.968267683``````

Bonus Tip: Visualize the data! Scatter plots and other graphs can make relationships between variables pop.

Remember: Interpreting regression output is an art, not a science. Use your domain knowledge, consider the context, and don’t hesitate to explore further!

So next time you face regression output, channel your inner R wizard and remember:

• Coefficients whisper about slopes and changes.
• P-values tell tales of significance, true or false.
• R-squared unveils the model’s explanatory magic.
• Residuals hold hidden clues, waiting to be discovered.

With these tools in your belt, you’ll be interpreting regression output like a pro in no time! Now go forth and conquer the data, fellow R adventurers!

Note: This is just a brief example. For a deeper dive, explore specific diagnostics, model selection techniques, and other advanced topics to truly master the art of regression interpretation.