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Last updated on 2024年8月5日

How do you interpret the results of a linear regression in R?

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Linear regression is a fundamental tool in data science for uncovering relationships between variables. When you run a linear regression in R, a programming language used for statistical computing, you receive a wealth of information that can guide your understanding of the data's behavior. Interpreting these results correctly is crucial for making informed decisions based on your analysis. Whether you're predicting outcomes, identifying trends, or simply exploring data relationships, knowing how to read the output of a linear regression in R is an essential skill.

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Nafees Iqbal

M.Sc. Data Science | AMU'26

1 Model Summary

The model summary in R provides a high-level overview of your linear regression results. It includes the R-squared value, which indicates the proportion of variance in the dependent variable that's explained by the independent variables. A value closer to 1 suggests a better fit. The Adjusted R-squared also accounts for the number of predictors in the model, making it a more reliable metric when comparing models with different numbers of predictors. The F-statistic and its corresponding p-value tell you if the overall regression model is statistically significant.

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Nafees Iqbal

M.Sc. Data Science | AMU'26
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In R programming we use the "lm" function to fit a regression line to data in the form of the formula of the linear model. In R we need a summary function to print the results of our fitted model and it contains all kinds of output. - The first line of the result gives the summary of the residuals (the distance of the residuals from zero). - The next line gives the least squares estimates (LSE) of the fitted line and p-value. Generally, we want the p-value for an independent variable to be less than 0.05. - The next line shows residual standard error. - The next line tells us about the multiple R2 and Adjusted R2. - The last line tells us about whether the R2 and Adj. R2 is statistically significant or not.

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2 Coefficients Table

The coefficients table is where you'll find the estimated effect of each independent variable on the dependent variable. Each coefficient represents the change in the dependent variable for a one-unit change in the independent variable, assuming all other variables are held constant. The p-values associated with each coefficient indicate whether these effects are statistically significant. If the p-value is below a certain threshold (commonly 0.05), you can reject the null hypothesis that there is no relationship between the variables.

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Nafees Iqbal

M.Sc. Data Science | AMU'26
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After fitting a regression model with the lm function in R and extracting the output via the summary() function, you will get a coefficient table that reveals the estimated effects of independent variables on the dependent variable. Each coefficient indicates the expected change in the dependent variable for a one-unit increase in the independent variable, while controlling for other variables. The p-values next to each coefficient assess statistical significance; a p-value below 0.05 typically allows you to reject the null hypothesis of no relationship. Individuals can also understand whether to reject the null hypothesis or not by the stars appearing besides each p-values, the more star there is the more likely it is to reject.

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3 Residuals Analysis

Residuals are the differences between observed values and those predicted by the model. Analyzing residuals can help you check the assumptions of linear regression, such as linearity, independence, and homoscedasticity (constant variance). In R, you can plot residuals against fitted values or independent variables to look for patterns. Ideally, residuals should be randomly scattered without discernible patterns, indicating that your model doesn't suffer from non-linearity or heteroscedasticity.

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Nafees Iqbal

M.Sc. Data Science | AMU'26
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Residual is the difference between the observed value and expected value. Residuals are crucial for assessing the validity of our model's assumptions. Ideally residuals should be independent and randomly distributed. Plotting the residuals against the fitted values or the independent variables can help us spot any patterns in the residuals. If the residuals are randomly scattered without any clear shape, it's a good sign that our model is capturing the relationships well and the assumptions are met. However if we see a curve or any definite shape in the residual plot, then it is an indication of Heteroscedasticity or non-linear relationship.

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4 Diagnostic Plots

R provides a set of diagnostic plots that help you visualize potential problems with your linear regression model. These include the Residuals vs Fitted plot, which can reveal non-linear relationships, the Normal Q-Q plot, which assesses whether residuals follow a normal distribution, and the Scale-Location plot, which checks for homoscedasticity. The Residuals vs Leverage plot helps you identify influential observations that might unduly affect the model's predictions.

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5 Leverage and Influence

Certain data points can have a disproportionate influence on your linear regression model. Leverage measures how far an independent variable deviates from its mean, and high-leverage points can distort your model. Influence combines leverage with the size of the residual to assess a point's impact on the regression coefficients. In R, you can use the influence.measures() function to identify such points, and consider removing them if they are also outliers or errors.

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6 Model Assumptions

Ensuring that your linear regression model meets the underlying assumptions is critical for valid results. These assumptions include linearity, independence of errors, homoscedasticity, and normality of residuals. In R, you can check these assumptions using various tests and plots, such as the Durbin-Watson test for independence, and the Shapiro-Wilk test for normality. If your model violates these assumptions, you may need to transform your data or consider a different modeling approach.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you interpret the results of a linear regression in R?

1

2

3

4

5

6

7

1 Model Summary

2 Coefficients Table

3 Residuals Analysis

4 Diagnostic Plots

5 Leverage and Influence

6 Model Assumptions

7 Here’s what else to consider

Data Science

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How do you interpret the results of a linear regression in R?

1

2

3

4

5

6

7

1 Model Summary

2 Coefficients Table

3 Residuals Analysis

4 Diagnostic Plots

5 Leverage and Influence

6 Model Assumptions

7 Here’s what else to consider

Data Science

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