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

**Steve's Data Tips and Tricks**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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# 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 (like for `wt`

!) means 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 Hornet Sportabout 1.006565971 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 Cadillac Fleetwood -1.634013415 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.

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