Influential Promotion Periods in a Sales Model

Why a Promotion Period Deserves a Closer Look

A sales period with unusually high advertising spend can strongly affect a fitted regression, even if its sales value is not far from the model’s prediction. That combination matters: the period may help determine the slope, seasonal contrast, and resulting predictions. The right response is to investigate how the period arose and how much conclusions depend on it, not to delete it merely because the fitted results change.

We will use a model with advertising spend and a seasonal indicator, then examine leverage and Cook’s distance. These diagnostics describe the role of a case in a particular fitted model; they do not establish that a data point is erroneous or that advertising caused higher sales.

A Sales Model and Its Conditions

Model: For sales period $i$, let $Y_i$ be sales revenue, $x_i$ be advertising spend, and $z_i$ be a seasonal contrast coded $-1$ for one season and $+1$ for the other. The multiple linear regression model is $$ Y_i=\beta_0+\beta_1x_i+\beta_2z_i+\varepsilon_i. $$ Here $\beta_0$ is the expected sales revenue at $x_i=0$ and $z_i=0$, a reference value between the two season codes; $\beta_1$ is the expected change in sales revenue per unit of advertising spend, holding season fixed; and $2\beta_2$ is the expected difference between the $+1$ and $-1$ seasons at the same advertising spend.

For the usual exact small-sample $t$- and $F$-based regression inference, assume the conditional mean follows this model, the errors have mean zero, are independent, and have a common variance $\sigma^2$; normal errors are additionally assumed for exact inference. The predictor values are treated as fixed for this analysis, and the design matrix must have full column rank. These conditions also clarify what the model does not guarantee: a seasonal indicator does not account for every time pattern, and independence may be questionable for successive sales periods.

Leverage depends on the predictors, not on the observed sales value. In matrix notation, $H=X(X^{\mathsf T}X)^{-1}X^{\mathsf T}$ is the hat matrix, and $h_{ii}$, its $i$th diagonal entry, is the leverage of period $i$. It measures how unusual that period’s predictor combination is relative to the other periods. For a model with $p$ fitted coefficients, the average leverage is $p/n$.

Influence concerns how much a fitted result changes when a case is omitted. Cook’s distance combines the case’s residual and leverage into a measure of the overall change in fitted values. It is a diagnostic, not a hypothesis test with a universal pass/fail cutoff.

Worked Example: Fitting Advertising and Season Predictors

Suppose ten hypothetical periods have advertising spend $x$ in thousands of dollars and sales revenue $Y$ in thousands of dollars. The seasonal code $z$ is $-1$ or $+1$. One $+1$ period is a promotion period with unusually high advertising spend.

Season code $z$	Advertising $x$	Sales $Y$	Period
+1	2	120	Ordinary
+1	3	125	Ordinary
+1	4	130	Ordinary
+1	5	135	Ordinary
+1	20	230	Promotion
-1	2	100	Ordinary
-1	3	105	Ordinary
-1	4	110	Ordinary
-1	5	115	Ordinary
-1	6	120	Ordinary

The predictor design has full rank: advertising varies within each season, and both seasonal codes occur. For a transparent calculation, the sales values in this constructed example follow $Y=100+5x+10z$, except that the promotion period has an additional 20 thousand dollars of sales. The primary model does not include a separate promotion indicator, so the promotion period can affect its estimates.

Within the $+1$ season, the mean advertising spend is $34/5=6.8$; within the $-1$ season it is $20/5=4$. The within-season sum of squares for advertising is $SSW=222.8+10=232.8$. After accounting for the season-specific means, the fitted advertising slope is the within-season covariance divided by $SSW$. The additional 20 in the promotion period contributes $20(20-6.8)=264$ to that covariance. Thus the slope increases from 5 to

\hat{\beta}_1=5+\frac{264}{232.8}=6.134.

The promotion’s contribution to the fitted $+1$-season intercept is $20/5-6.8(264/232.8)=-3.711$. Its contribution to the $-1$-season intercept is $-4(264/232.8)=-4.536$. Converting these two intercept changes to the common intercept and seasonal contrast gives the fitted model, rounded to three decimal places:

\hat{Y}=95.876+6.134x+10.412z.

For the promotion period, the fitted sales value is approximately $228.969$ thousand dollars, using the unrounded fitted coefficients. Its residual is $230-228.969=1.031$ thousand dollars. That residual is not especially large relative to the residual standard error calculated below. As the next calculation shows, the period can nevertheless be highly influential because its advertising spend is unusual.

Why the Promotion Period Has High Leverage

In this balanced seasonal model, the fitted values can also be represented by separate intercepts for the two seasons and one shared advertising slope. The leverage of a period in season group $g$ is therefore

h_{ii}=\frac{1}{n_g}+\frac{(x_i-\bar{x}_g)^2}{SSW},

where $n_g$ is the number of periods in that season, $\bar{x}_g$ is its mean advertising spend, and $SSW$ is the sum of squared within-season advertising deviations. The first term accounts for estimating that season’s intercept; the second accounts for the shared slope.

Worked Example: Calculating the Promotion Period’s Leverage

There are five periods in each season. For the promotion period, $x_i=20$, the $+1$-season mean is $\bar{x}_g=6.8$, and $SSW=232.8$. Therefore,

h_{ii}=\frac{1}{5}+\frac{(20-6.8)^2}{232.8} =0.2+\frac{174.24}{232.8} =0.94845.

This leverage is much larger than the model’s average leverage, $p/n=3/10=0.3$. A sometimes-used screening guideline flags leverage above $2p/n=0.6$; this period exceeds that guideline, but the guideline is not a formal decision rule. The more important point is substantive: advertising spend of 20 is far from the other $+1$-season values, so this period has unusual predictor values and substantial potential to affect the fitted line.

A high-leverage case need not be influential if its observed outcome agrees with the pattern supported by the other cases. Conversely, a case’s residual alone does not tell the whole story. Diagnostics should therefore examine both predictor unusualness and changes in the fitted results.

Cook’s Distance and a Sensitivity Check

For ordinary least squares with $p$ fitted coefficients, residual $e_i=Y_i-\hat{Y}_i$, residual mean square $MSE$, and leverage $h_{ii}$, Cook’s distance is

D_i=\frac{e_i^2}{p\,MSE}\frac{h_{ii}}{(1-h_{ii})^2}.

It summarizes how much the fitted values would change if case $i$ were omitted and the model refitted. Large values merit investigation in context. Since cutoffs are only rough screening aids, report the diagnostic alongside the coefficient or prediction changes that matter for the analysis.

Worked Example: Computing Cook’s Distance

The model has $n=10$, $p=3$, and residual degrees of freedom $df=n-p=7$. For the constructed data, the promotion period’s residual is $e_i=100/97\approx1.031$, and its leverage is $h_{ii}=92/97$. The residual sum of squares for this one-period perturbation is $20^2(1-h_{ii})=2000/97\approx20.619$, so

MSE=\frac{2000/97}{7}\approx2.946,\qquad D_i=\frac{(100/97)^2}{3((2000/97)/7)} \frac{92/97}{(1-92/97)^2} =\frac{644}{15}\approx42.93.

This is a large Cook’s distance: the period has a substantial effect on the fitted values. Notice that its residual standard error is $\sqrt{2.946}\approx1.716$ thousand dollars, so its residual is only about $1.031/1.716\approx0.601$ residual standard errors. The example shows why a modest residual does not make a high-leverage case harmless.

Comparing Fits Without Hiding the Promotion

Worked Example: Interpreting a Deletion Fit as a Diagnostic

Refit the advertising-and-season model after temporarily omitting the promotion period. The remaining nine values follow $Y=100+5x+10z$ exactly, so the deletion-fit coefficients are an intercept of 100, an advertising slope of 5, and a seasonal coefficient of 10. Compared with the full-data estimates, the deletion-fit intercept is higher by about 4.124 thousand dollars, its slope is lower by about 1.134 thousand dollars of sales per thousand dollars of advertising, and its seasonal coefficient is lower by about 0.412 thousand dollars per unit of the $-1,+1$ contrast.

This comparison establishes that the full-data estimates are sensitive to the promotion period. It does not establish that the deletion fit is the correct answer. The period is a valid observation, and the deletion fit represents a narrower data set. In this deliberately constructed example, the remaining data have zero residual error, which is an artificial consequence of how the values were chosen; it should not be taken as evidence of perfect prediction or precise uncertainty in a real sales process.

Worked Example: A Model That Represents Promotion Status

Suppose promotion status is known to be important for the intended analysis. A sensitivity model can include an indicator $P_i$, equal to 1 for a promotion period and 0 otherwise:

Y_i=\beta_0+\beta_1x_i+\beta_2z_i+\beta_3P_i+\varepsilon_i.

For the constructed data, the coefficients are exactly $\hat{\beta}_0=100$, $\hat{\beta}_1=5$, $\hat{\beta}_2=10$, and $\hat{\beta}_3=20$, in thousands of dollars for the relevant outcome or contrast. Unlike the deletion fit, this model retains all ten periods and represents the promotion period’s additional sales with a separate term. In actual data, the coefficients would not generally be exact, and the extra term would need substantive justification. With only one promotion period here, it also cannot establish that other promotions would have the same effect.

The primary model, the deletion fit, and a promotion-indicator model answer different questions. The first describes the specified advertising-and-season relationship across all periods; the deletion fit describes a data set that excludes the promotion; the indicator model allows a distinct promotion-period difference. A useful sensitivity analysis explains these differences rather than presenting one fit as a corrected version of another.

Investigate Before You Interpret

A large influence diagnostic is a prompt to check the data and model, not a command to remove a case. Confirm that advertising spend and sales are recorded correctly, and ask whether the period reflects a genuine campaign, a change in sales measurement, or an unusual event that the model omits. Then decide whether the model and its target question are still appropriate.

If the period is valid, retain it in the primary analysis unless the stated target population or question explicitly excludes promotion periods. Report how important estimates change across scientifically justified fits, and explain what each fit represents. Do not choose a fit simply because it produces a more convenient slope or smaller residuals. Because the observations are sales periods, also consider whether a trend or dependence between successive periods could undermine the model assumptions; the seasonal indicator alone does not resolve those issues.

Common mistake: “Cook’s distance is large, so the promotion period is an outlier and should be deleted.”
Correction: Cook’s distance indicates sensitivity of the fit, not invalidity of the observation. Check the record, the context, the model, and the target question; preserve valid observations and make sensitivity comparisons explicit.

Key takeaway: Leverage identifies unusual predictor combinations; Cook’s distance combines leverage with residual information to flag influential cases. Use these diagnostics to understand and communicate sensitivity, not to remove valid observations merely to improve a result.

Check Your Understanding

Use the model and examples above to answer these questions.

In the seasonal contrast coding, what does $2\beta_2$ represent when advertising spend is held fixed?
Why can a period with a modest residual still have a large Cook’s distance?
What is the promotion period’s leverage in the worked example, and what feature of its predictors explains it?
What different question does the deletion fit answer compared with the full-data model?
Give one reason a promotion indicator might be justified and one limitation of estimating its effect from only one promotion period.