Regression Analysis: How to DO It Example: The “car discount” dataset

Regression Analysis:How to DO It

Example: The “car discount” dataset

Discounts on Car Purchases

• Of course, no one pays list price for a new car. Realizing this, the owner of a new-car dealership has decided to conduct a study, to attempt to understand better the relationship between customer characteristics, and customer success in negotiating a discount from his salespeople.

• He collects data on a sample of 100 purchasers of mid-size cars (he has already sold several thousand of these cars):– Specifically, he notes the age, annual income, and sex (men

were represented by 0, and women by 1, in the coding of sex) of each purchaser (obtained from credit records), together with the discount from list price which the purchaser finally received.

Discounts on Car Purchases

• He collects data on a sample of 100 purchasers of mid-size cars (he has already sold several thousand of these cars):– He notes the age, annual income, and sex of each purchaser,

together with the discount from list price which the purchaser finally received.

Discount Age Income Sex1003 28 47658 11394 41 32126 1 Discount ($) negotiated on the purchase2542 21 28374 1 of a car: age of purchaser (years),1658 47 29321 0 annual income ($), and sex (M/F = 0/1).1374 29 38016 11536 43 25343 01402 54 30310 0692 35 45709 0947 41 46242 0

1415 19 27933 1… … … …

Discounts on Car Purchases• He collects data on a sample of 100 purchasers of mid-size cars (he

has already sold several thousand of these cars):– He notes the age, annual income, and sex of each purchaser, together

with the discount from list price which the purchaser finally received.• Why mid-size cars only?

– To avoid needing to include model/price of car• Other possible explanatory variables?

– About purchaser• Negotiation training• Preparatory research• Significant other

– About salesperson• Identity• Biases

Look at the Univariate Statistics

• This will give you a sense of how each variable varies individually– Estimate of population mean (or proportion)– Standard deviation and extremes– 95%-confidence interval for population mean (or proportion)

• Estimate ± (~2)·(standard error of the mean)• Estimate ± “margin of error” (at 95%-confidence level)

Univariate statisticsDiscount Age Income Sex

mean 1268.24 37.1 35705.17 0.46standard deviation 538.665375 9.91122209 10273.7291 0.50090827standard error of the mean 53.8665375 0.99112221 1027.37291 0.05009083

minimum 130 19 19119 0median 1310.5 37 34401.5 0maximum 2542 58 64648 1range 2412 39 45529 1

skewness -0.018 0.154 0.452 0.163kurtosis -0.710 -0.633 -0.270 -2.014

number of observations 100

t-statistic for computing95%-confidence intervals 1.9842

For example, $1,268.24 ± 1.9842·$53.87, or 46% ± 1.9842·5.01% .

The Full Regression

• The “most-complete” model provides …– The best predictive model (pretty much)– The most accurate estimate of the “pure effect” of

each explanatory variable on the dependent variable• Specifically, the difference in the dependent variable

typically associated with one unit of difference in one explanatory variable when the others are held constant.

Regression: Discountconstant Age Income Sex

coefficient 1971.72565 9.48991379 -0.035313 446.294355std error of coef 146.147064 3.6320188 0.00366827 64.5567912t-ratio 13.4914 2.6128 -9.6266 6.9132significance 0.0000% 1.0423% 0.0000% 0.0000%beta-weight 0.1746 -0.6735 0.4150

standard error of regression 301.19175coefficient of determination 69.68%adjusted coef of determination 68.74%

number of observations 100residual degrees of freedom 96


The Adjusted Coefficient of Determination in the Full Model

• How much of the “story” (how much of the overall variation in the dependent variable) is potentially explained by the fact that the explanatory variables themselves vary across the population?

• r2 = 1 – Var() / Var(Y) (roughly) = 68.74%

– How can it be increased?• By including new relevant variables• Including a new “garbage” variable will leave it, on

average, unchanged

The Coefficients

• The coefficient of an explanatory variable in the most-complete model …– Is an estimate of the average difference in the

dependent variable for two distinct individuals who differ (by one unit) only in that explanatory variable.

– Is an estimate of the average difference we’d expect to see in a specific individual if one aspect alone were slightly different (and all other aspects were the same.)

• coefficient ± (~2)·(standard error of coefficient)






$9.49 ± 1.9850·$3.63 per year of Age (with same Income and Sex), or-$0.0353 ± 1.9850·$0.0037 per dollar of Income (with same Age and Sex), or$446.29 ± 1.9850·$64.56 more for a woman (1) than for a man (0) (with same Age and Income)

Predictions, using most-recent regression

coefficients values for predictionconstant 1971.7256

Age 9.4899138 30 31 30 30Income -0.035313 35000 35000 36000 35000

Sex 446.29435 1 1 1 0

predicted value of Discount 1466.762 1476.252 1431.449 1020.468standard error of prediction 305.5644 305.2995 305.8382 305.2382standard error of regression 301.1917 301.1917 301.1917 301.1917standard error of estimated mean 51.50843 49.91316 53.10864 49.53689

confidence level 95.00% t-statistic 1.9850residual degr. freedom 96

confidence limits lower 860.2218 870.2374 824.3652 414.5748for prediction upper 2073.303 2082.267 2038.533 1626.361

confidence limits lower 1364.519 1377.175 1326.029 922.1379for estimated mean upper 1569.006 1575.329 1536.869 1118.798

Predict

Make single prediction

Tests involving Coefficients• In the full model, how strongly does the evidence

support saying, “Sex≥$200”?• H0: Sex≤$200, significance 0.01204% (overwhelmingly strong

evidence against H0, hence supporting original statement)

446.294 estimate/prediction of unknown quantity64.557 measure of uncertainty

100 sample size3 number of explanatory variables in regression, or

0 if dealing with a population mean

Null hypothesis:≥ 100.00000%

true value = 200 0.02408%≤ 0.01204%

(from t-distribution with 96 degrees of freedom)

significance level of data with respect to null hypothesis

From Session-1’s “Hypothesis_Testing_Tool.xls”

Tests involving Coefficients• Other statements?

Statement Significance level of data (with respect to opposite statement)

Strength of evidence supporting statement

Sex≥$200 0.01204% overwhelming

Sex≥$300 1.28444% very strong

Sex≥$350 6.95385% somewhat strong

Sex≥$400 23.75235% quite weak

From Session-1’s “Hypothesis_Testing_Tool.xls”

Predictions

• Based on ANY model, what would we predict the dependent variable to be, if all we knew about an individual were the given values for the listed explanatory variables?

• Prediction ± (~2)·(standard error of the prediction)

• What would we expect to see, on average, across a large pool of similar individuals?

• Prediction ± (~2)·(std. error of the estimated mean)

Prediction, using most-recent regression

constant Age Income Sexcoefficients 1971.726 9.489914 -0.03531 446.2944values for prediction 30 35000 1

predicted value of Discount 1466.762standard error of prediction 305.5644standard error of regression 301.1917standard error of estimated mean 51.50843

confidence level 95.00% t-statistic 1.9850residual degr. freedom 96

confidence limits lower 860.2218for prediction upper 2073.303

confidence limits lower 1364.519for estimated mean upper 1569.006

Predict

Make multiple predictions

$1,466.76 ± 1.9850·$305.56, an individual predictionfor a 30-year-old woman earning $35,000/year

$1,466.76 ± 1.9850·$51.51, an estimate of the large-group meanfor 30-year-old women earning $35,000/year

Significance

• The significance level of the t-ratio (for each variable separately)

• Sometimes called the “p-value” for that variable

– How strong is the evidence that, in a model already containing all of the other explanatory variables, this variable “belongs” (i.e., has a non-zero coefficient of its own)?

– Equivalently, is this a variable whose value we’d like to know when predicting for a specific individual?

• Close to zero = strong evidence it DOES belong (our null hypothesis is that it doesn’t)






Significance (continued)

• Null hypothesis: “In the current model, the true coefficient of this variable is 0.”– The coefficient of this variable is our estimate– (coefficient) / (standard error of the coefficient) tells us how many standard deviations away from the hypothesized truth (0) the estimate is

• significance = Pr(we’d be this far away just by chance)

– Close to 0% = (recall coin-flipping story)• highly contradictory to null hypothesis• strongly supportive of alternative (it DOES belong)

Significance (continued)

• The significance level deals with the marginal contribution of a variable to the current model.

• Adding an irrelevant explanatory variable to a regression model will increase the adjusted coefficient of determination about half the time. The significance level tells us if the coefficient of determination went up by enough to argue that the new variable is relevant.

The Beta-Weights

• Why is Discount varying from one sale to the next?– What’s the relative explanatory “power” of

(variation in) each of the explanatory variables (in explaining the currently-observed variability in the dependent variable across the population)?

– The comparative magnitudes of the beta-weights (for all of the explanatory variables together in the model) answer this question.






• Why does discount vary across the population?

• Primarily, because Income varies.• Secondarily, because some purchasers are men and

others are women (i.e., Sex varies).

The Beta-Weights (continued)

• Each answers the question:– If two individuals have the same values for all the

explanatory variables in the model except one, and for this one their values differ by one standard-deviation’s-worth of variability (in this variable), then their predicted values for the dependent variable would differ by how many standard deviations (of variability in the dependent variable)?

• “Typical” variation in each of the explanatory variables alone can explain (relatively) how much of the observed variability in the dependent variable?

We Can Explore Other Models

• We can drop variables– Are older or younger purchasers currently getting

larger discounts?• We can change the dependent variable

– Are the female purchasers, on average, older or younger than the male purchasers?

– What’s the impact of aging on purchaser income?

Regression: Ageconstant Sex

coefficient 38.9074074 -3.9291465std error of coef 1.32861376 1.95893412t-ratio 29.2842 -2.0058significance 0.0000% 4.7639%beta-weight -0.1986




Male purchasers are, on average, 38.91 years old.

Female purchasers are, on average, 3.93 years younger than the men.

Are the female purchasers, on average, older or younger than the male purchasers?

Regression: Discountconstant Age

coefficient 1817.16511 -14.795825std error of coef 202.794627 5.28272248t-ratio 8.9606 -2.8008significance 0.0000% 0.6142%beta-weight -0.2722




If the “pure” effect of an additional year of age is to increase a purchaser’s discount, then what explains the negative coefficient of Age below?

• An older patron is likely to have a higher income (which typically is associated with a smaller discount)

• An older patron is more likely to be male (which typically is associated with a smaller discount)

Regression: Discountconstant Age Sex

coefficient 1292.48764 -8.4694585 630.367989std error of coef 178.496906 4.34612096 85.9945283t-ratio 7.2410 -1.9487 7.3303significance 0.0000% 5.4216% 0.0000%beta-weight -0.1558 0.5862

A Reconciliation across Models

• On these next three slides, we’ll focus on the “older people have higher incomes” effect:

• As a patron ages by a year (and his/her sex stays unchanged!), his/her discount typically drops by $8.47.

As the patron ages by a year (and his/her sex stays unchanged!), his/her income typically rises by $508.58.

Regression: Incomeconstant Age Sex

coefficient 19234.7835 508.57672 -5212.6301std error of coef 3542.55077 86.25558 1706.69615t-ratio 5.4296 5.8962 -3.0542significance 0.0000% 0.0000% 0.2913%beta-weight 0.4906 -0.2541

The combined age and income effects are precisely what we originally estimated for an additional year of age, when income was not held constant.



9.48991379 impact of Age508.57672 additional Income

-17.959372 impact of additional Income

net consequence of-8.4694585 aging a year and earning

more as a result

Conclusion

Regression: Discountconstant Age Sex

coefficient 1292.48764 -8.4694585 630.367989std error of coef 178.496906 4.34612096 85.9945283t-ratio 7.2410 -1.9487 7.3303significance 0.0000% 5.4216% 0.0000%beta-weight -0.1558 0.5862

To the extent that Income covaries with Age, if Income is omitted from our model, Age gets “blamed” for part of Income’s effect on Discount.

This yields the most accurate possible predictions based on Age and Sex alone, but grossly misestimates the pure effect of Age.

And that is why we try to use the “most-complete” model to estimate the pure effect of any variable on the dependent variable… and why our next session will focus on building the model itself.

Summary: Questions a Regression Study can Answer

Make an Individual PredictionPredict a variable (with an unknown value) for an individual, given some specific information about that individual.• Regress the variable-to-be-predicted (the dependent variable) onto

the known variables (the independent or explanatory variables), and make a prediction.

• The margin of error in the prediction is (~2) (the standard error of ∙the prediction).

Example: “Predict the discount from list price that a 30-year-old woman who buys an intermediate-sized vehicle from the dealership would receive.”

$1668.77 ± 1.9847 $420.06∙

Estimate a Group MeanEstimate the mean value of a variable, across a (large) group of individuals who share certain specific characteristics.• Regress the first variable onto the others. Then make a prediction

of the variable for one of the individuals (which will be used as the estimate of the mean across this group of similar individuals).

• The margin of error in the estimated mean is (~2) (the standard ∙error of the estimated mean).

Example: “Estimate the mean discount received by 30-year-old women (plural!) who buy intermediate-sized vehicles from the dealership.”

$1668.77 ± 1.9847 $65.60∙

Estimate a “Pure” Difference (1)What is the mean difference in the value of the dependent variable typically associated with a one-unit difference in another variable, when everything else of relevance remains unchanged?• Regress the dependent variable onto all of the other variables in the study (the

“most complete” model), and look at the coefficient of the “other” variable.• The margin of error in the estimated mean associated difference is (~2) (the ∙

standard error of the coefficient).Example: What is the average difference in negotiated discount associated with an incremental year of age of the purchaser of an intermediate-sized car from the dealership, when all other characteristics of that purchaser remain unchanged?

$9.49 ± 1.9850 $3.63∙

Estimate a “Pure” Effect (2)

Example: What is the average difference in negotiated discount associated with an incremental year of age of the purchaser of an intermediate-sized car from the dealership, when all other characteristics of that purchaser remain unchanged?Example: What is the average effect of an incremental year of age on negotiated discount?If you’re willing to assert that the linkage between age and negotiated discount is causal (we’ll discuss “causality” in our next class), then the “average pure difference” and “average pure effect” questions can be viewed as the same.

Estimate a Confounded Difference (1)

What is the mean difference in the value of the dependent variable typically associated with a one-unit difference in another variable, when all remaining variables consequently may take different values themselves?• Regress the dependent variable onto just the one variable, and look at

the coefficient of the explanatory variable.• The margin of error in the estimated mean difference is (~2) (the ∙

standard error of the coefficient).Example: As 30-year-old purchasers age by a year, estimate the average change in their negotiated discounts.

$-14.80 ± 1.9845 $5.28 ∙

Estimate a Confounded Difference (2)

Example (continued): As 30-year-old purchasers age by a year, estimate the average change in their negotiated discounts. The older purchasers would, on average receive smaller discounts. This is because, as Age increases for purchasers, Income tends to increase as well. The additional Age increases Discount, the additional Income tends to decrease Discount, and the net effect just happens to be a decrease.

Measure the Potential Explanatory Power of a Model

How much of the variation in the dependent variable is potentially explained by the fact that several explanatory variables vary from one individual to the next?• Regress the first variable (the dependent variable) onto the

other variables (the independent or explanatory variables), and look at the adjusted coefficient of determination.

Example: “How much of the variation in negotiated Discounts on intermediate-size cars can be potentially explained by the facts that Age, Income, and Sex all vary from one purchaser to the next?”

68.74%

Rank the Explanatory Variables by Relative Explanatory Importance

When all the variables are considered together, typical variation in which would lead to the greatest expected variation in the dependent variable.• Regress the dependent variable-to-be-predicted (the dependent

variable) onto the explanatory variable. • Compare the magnitudes (absolute values) of the beta-weights of

the explanatory variables.Example: “Why does Discount vary from one purchaser to the next?”“Because Income varies (-0.6735). And secondarily, because Sex varies (some are men, and others women) (0.4150).”

Evaluate a Variable’s Model Inclusion (1)

Given a particular regression model, how strong is the (supporting) evidence that a specific one of the explanatory variables has a true non-zero effect on the dependent variable (and therefore "belongs" in the model)?• To see if evidence supports a claim, we always take the opposite as the null hypothesis: in

this case, to say that a variable does not belong in the model we say “H0: coefficient (of the explanatory variable) = 0.”

• The displayed significance level for that variable is with respect to the “doesn’t belong” null hypothesis, so a large numeric significance level indicates little or no evidence that the variable belongs in the model.

• However, a small significance level provides strong evidence against the null hypothesis, and therefore strong evidence that the explanatory variable plays a non-zero role in the relationship.

Evaluate a Variable’s Model Inclusion (2)Example:



We see here overwhelmingly-strong evidence that Income and Sex have non-zero effects and “belong” in our model, and very strong evidence that Age belongs as well.

Documents

Regression Analysis: How to DO It Example: The “car discount” dataset