The multi-channel advertising model explains 91.5% of sales variance (R² = 0.9149, Adjusted R² = 0.9136), with an overall fit that is statistically significant (F = 702.1, p = 0). The channel with the largest standardized effect on sales is tv_spend, meaning a one-standard-deviation increase in that channel's budget produces the greatest lift relative to its own variability. Prediction error is 1.49 units of sales on average, giving decision-makers a reliable baseline for budget allocation.

The box plot shows the spend distribution for each advertising channel across 200 budget periods. Medians per channel: tv_spend (med= 164.9 ); radio_spend (med= 22.4 ); newspaper_spend (med= 52.8 ). Channels with wider interquartile ranges have more variable budgets, which gives the regression more statistical power to estimate their effect. Outlier periods (extreme spends) can exert undue influence on coefficients and should be reviewed before finalising budget recommendations.

The heatmap shows pairwise Pearson correlations between all spend channels and sales. The channel most strongly correlated with sales is tv_spend (r = 0.757). Correlations between spend channels reveal multicollinearity risk: values above 0.7 indicate that two channels move together in the budget, which can inflate standard errors and make individual coefficient estimates less reliable.

Standardized coefficients (beta weights) measure each channel's effect in standard-deviation units, making them directly comparable regardless of how dollars are scaled. The dominant channel by standardized effect is tv_spend (β = 0.742). Channels with beta weights near zero contribute negligible independent lift to sales after accounting for the other channels in the model.

The regression table shows raw coefficients (sales change per unit of spend), standard errors, t-statistics, p-values, and 95% confidence intervals for each channel. 2 of 3 channels are statistically significant (p < 0.05). The largest raw effect is radio_spend (coefficient = 0.2051), meaning each additional unit of radio_spend spend is associated with 0.2051 additional sales units, holding other channels constant.

Each point compares the model's predicted sales to the actual observed sales for one budget period. Points on the 45-degree diagonal indicate perfect predictions; scatter around the line represents unexplained variation. The model achieves R² = 91.5%, explaining 91.5% of total sales variance, with a root mean squared error of 1.49 units. Systematic curves or funnels in this chart would suggest the linear model needs transformation or additional variables.

Residuals (actual minus predicted) plotted against fitted values check whether the OLS assumptions hold. A random horizontal band around zero indicates valid homoscedasticity (constant variance). Mean residual = 0 (should be near 0); maximum absolute residual = 4.43. A funnel shape (widening variance at higher fitted values) would suggest log-transforming the sales outcome, while a curve would indicate a non-linear relationship that a polynomial or interaction term might address.

ROI estimates are the raw regression coefficients: sales units generated per additional unit of spend in each channel, holding all other channels constant. The highest-ROI channel is radio_spend with an estimated 0.2051 sales units per unit of spend. Note that these estimates assume the current spend mix and may not hold at dramatically different budget levels due to diminishing marginal returns — channels ranking near zero or negative are candidates for budget reallocation.