Exploring Two–Variable Data

Stratified sampling

Cluster sampling

Simple random sampling

Multi-stage sampling

Observational studies do not control for lurking variables which could influence both explanatory and response variables, potentially inflating $r^2$ without causal linkages.

Correlation coefficients derived from observational data typically underestimate true cause-and-effect relationships leading to deflated $r^2$ values even when causation exists.

The computation method for $r^2$ inherently includes confounding factors that always imply direct causation when high values are observed in observational studies.

Controlled experiments do not utilize $r^2$ because it's only applicable in non-experimental contexts like observational studies where causation can be inferred directly from it.

A clear curved pattern.

Points equally distributed above and below the x-axis.

No distinct patterns or trends in residuals across different values of x.

A random scatter around zero.

An r-squared comparison alone determines necessity cubic inclusion without further testing.

Linear correlation coefficient increases sufficiently when adding cubic term compared prior version.

A partial F-test comparing full cubic model against reduced quadratic model could show improvement in fit.

Simple t-tests on coefficients for individual cubic terms would suffice independent assessment context.

A completely random pattern showing no identifiable trend or structure.

Clear clustering around certain values on the x-axis without any apparent curve.

A consistent horizontal banding suggesting perfectly equal variance among errors.

A curved pattern indicating that relationship between variables may be non-linear.

Assuming direct cause-and-effect based solely on model fit.

Looking for confounding variables that could explain any observed association.

Accepting causation because high coefficient determination ($r^2$).

Relying purely on p-values without considering practical significance.

Residuals spread equally above and below the horizontal axis at all values of x.

Randomly distributed residuals around the horizontal axis without any apparent pattern.

Constant variance of residuals as x increases.

A systematic pattern in the residual plot.

Small sample sizes might not detect small but meaningful effects, while large sample sizes could find significant results for effects that aren't truly important.

Effect size has no bearing on study design as sample size alone can ensure valid statistical conclusions are drawn.

Focusing solely on effect size while ignoring sample size is recommended since it is the only measure needed to prove a strong relationship.

Large sample sizes always increase the chance of finding true positive relationships, even for true negative or weak effect sizes.

The lack of pattern in the residuals confirms a good fit of the linear model.

The model does not adequately capture the relationship between the variables.

A transformation of variables is unnecessary since patterns in residuals are common.

Adding more data points will eliminate any patterns seen in the residuals.

Points that improve the overall fit of the regression model.

Points with x-values that are far away from the rest of the data.

Points with low-magnitude residuals.

Points with y-values that are far away from the rest of the data.