Regression Analysis: Joins, Inference, Diagnostics, and Prediction

Why do we perform EDA before regression?

To understand the data and check if a regression model makes sense.

What is the difference between a left join and an inner join?

A left join keeps every row from the left table. An inner join keeps only rows that have a match in both tables.

Why can one channel's subscriber count appear many times after a join?

One channel can have many videos, so the channel's information is repeated for each video.

What happens if there is no matching key in a left join?

The row stays, but the missing information from the other table becomes NA.

What does correlation measure?

The strength and direction of a linear relationship between two variables.

Does correlation imply causation?

No. Two variables can be related without one causing the other.

Can correlation near 0 occur with a nonlinear relationship?

Yes. Correlation measures linear relationships, so it can miss nonlinear relationships.

Why might log transformations help skewed variables?

They can reduce skew and make patterns easier to see and model.

What does the error term represent?

The part of Y that the model does not explain.

Why can't we perfectly predict Y from X?

Other factors affect Y, creating variation that the model cannot explain.

What is the difference between β₁ and β̂₁?

β₁ is the true population slope. β̂₁ is our estimate of the slope from the sample.

What does the slope mean?

The expected change in Y for a 1-unit increase in X.

What does the intercept mean?

The expected value of Y when X = 0.

Why does least squares square residuals?

So positive and negative errors cannot cancel each other out.

Why doesn't the sample slope automatically tell us the true population slope?

A sample is only part of the population, so the estimate has sampling variability.

What does H₀: β₁ = 0 mean?

There is no linear relationship between X and Y in the population.

What does a small p-value tell us?

The data would be unusual if the null hypothesis were true.

What does a p-value NOT tell us?

It does not tell us the probability that the null hypothesis is true.

What does a confidence interval tell us?

It gives a range of plausible values for the population parameter.

How does a CI relate to a hypothesis test?

If a 95% CI does not include 0, that agrees with rejecting H₀: β = 0 at about the 5% level.

Why do we need inference if we already have an estimated slope?

Inference tells us how much uncertainty there is around our estimated slope.

Why can R give you a p-value you shouldn't trust?

If the model assumptions fail, the p-value may not be reliable.

What does LINE stand for?

Linearity, Independence, Normality, Equal spread.

How can a residual plot show nonlinearity?

A curved or patterned shape suggests the relationship is not linear.

What does a funnel-shaped residual plot suggest?

The spread changes, so the equal-spread assumption may be violated.

Why is independence related to data collection?

Observations may be related because of how or when they were collected.

Are we checking normality of X, Y, or residuals?

Residuals.

Why do model assumptions matter for inference?

Inference methods rely on assumptions. If they are violated, the results can be unreliable.

What does R² measure?

The proportion of variation in Y explained by the model.

Why doesn't high R² prove causation?

A model can explain variation without showing that X causes Y.

Why use multiple regression instead of several separate simple regressions?

It lets us study several predictors together and account for their relationships.

What does 'holding all other variables constant' mean?

We look at the effect of one predictor while keeping the other predictors the same.

Why can a coefficient change when another predictor is added?

The predictors may overlap in what they explain about Y.

What is confounding?

When predictor effects become mixed together because predictors are related to each other.

What is collinearity?

When predictors are highly related, making their individual effects harder to separate.

Why can a predictor be significant in one model but not another?

Adding predictors can change the estimated effect and its uncertainty.

What does H₀: βⱼ = 0 mean in MLR?

That predictor has no linear relationship with Y after accounting for the other predictors.

Why is interpreting an MLR coefficient different from an SLR coefficient?

An MLR coefficient describes the relationship while holding the other predictors constant.

What is the difference between statistical and practical significance?

Statistical significance asks whether there is evidence an effect exists. Practical significance asks whether the effect is large enough to matter.

Why aren't regression predictions exact?

The model has uncertainty, and individual observations have random variation.

What is the difference between predicting the mean and predicting a new observation?

A mean prediction estimates the average Y. A new-observation prediction estimates one individual Y.

Why is a prediction interval wider than a confidence interval?

It includes both model uncertainty and individual-to-individual variation.

Why can in-sample performance be misleading?

The model was trained on those same observations, so its performance can look too good.

What is the purpose of cross-validation?

To estimate how well a model will predict new, unseen data.

What happens in one iteration of K-fold cross-validation?

Train on K−1 folds and test on the remaining fold.

Why is out-of-sample performance more useful?

It better shows how the model will perform on new data.

What does RMSE tell you?

The typical size of prediction errors, measured in the units of Y.

Why is RMSE easier to interpret than MSE?

RMSE is in the original units of Y, while MSE is in squared units.

If Model A has a lower RMSE than Model B, what does that tell you?

Model A has smaller typical prediction errors and better predictive performance.