A problem of interest to health officials (and others) is to...

A problem of interest to health officials (and others) is to determine the effects of smoking during pregnancy on infant health. One measure of infant health is birth weight; a birth weight that is too low can put an infant at risk for contracting various illnesses. Since factors other than cigarette smoking that affect birth weight are likely to be correlated with smoking, we should take those factors into account. For example, higher income generally results in access to better prenatal care, as well as better nutrition for the mother. An equation that recognizes this is $$=\beta_{0}+\beta_{1} \text { cigs }+\beta_{2} \text { faminc }+u$$ (i) What is the most likely sign for $\beta_{2} ?$ (ii) Do you think cigs and faminc are likely to be correlated? Explain why the correlation might be positive or negative. (iii) Now, estimate the equation with and without faminc, using the data in BWGHT. Report the re- sults in equation form, including the sample size and $R$ -squared. Discuss your results, focusing on whether adding faminc substantially changes the estimated effect of cigs on bwght.

A problem of interest to health officials (and others) is to determine the effects of smoking during
pregnancy on infant health. One measure of infant health is birth weight; a birth weight that is too low can put an infant at risk for contracting various illnesses. Since factors other than cigarette smoking that affect birth weight are likely to be correlated with smoking, we should take those factors into account. For example, higher income generally results in access to better prenatal care, as well as better nutrition for the mother. An equation that recognizes this is
$$=\beta_{0}+\beta_{1} \text { cigs }+\beta_{2} \text { faminc }+u$$
(i) What is the most likely sign for $\beta_{2} ?$
(ii) Do you think cigs and faminc are likely to be correlated? Explain why the correlation might be
positive or negative.
(iii) Now, estimate the equation with and without faminc, using the data in BWGHT. Report the re-
sults in equation form, including the sample size and $R$ -squared. Discuss your results, focusing
on whether adding faminc substantially changes the estimated effect of cigs on bwght.

Key Concepts

Control Variables

Control variables are additional independent variables included in a regression model to account for other factors that might influence the dependent variable. Including control variables like family income in the infant birth weight study helps isolate the specific effect of smoking by controlling for other determinants of birth weight, reducing potential confounding and thereby increasing the reliability of the conclusions drawn from the analysis.

Correlation Among Explanatory Variables

In multiple regression, explanatory variables may be correlated with one another. This correlation, known as multicollinearity, can complicate the interpretation of individual coefficients because it is challenging to disentangle each variable's unique effect on the dependent variable. In the scenario, if family income and cigarette use are correlated (for instance, higher income might be associated with lower smoking frequency), their coefficients might partially capture each other’s effects.

Coefficient Interpretation

Coefficient interpretation in regression analysis involves understanding the expected change in the dependent variable for a one-unit change in an independent variable, holding other variables constant. For instance, a positive coefficient on family income would mean that, all else being equal, higher family income is associated with higher birth weight. This allows researchers to quantify the influence of each factor on the outcome.

Multiple Regression Analysis

Multiple regression analysis is a statistical method used to model the relationship between a dependent variable and several independent variables. This technique allows researchers to assess the contribution of each explanatory variable in predicting the outcome while controlling for the influence of other factors in the model. This is especially useful when factors that can affect the dependent variable are interrelated, ensuring that the estimated effects are not biased by omitted factors.

Omitted Variable Bias

Omitted variable bias occurs when a model leaves out one or more important explanatory variables that are correlated with both the independent variable of interest and the dependent variable. This exclusion can lead to biased and inconsistent estimates of the coefficients, misleading the interpretation of the relationship. In the context described, not accounting for factors like family income that affect infant birth weight might distort the true effect of smoking on the outcome.

Transcript

00:01 Hi everyone.

00:03 We have this computer exercise on chapter 3 and the problem is to determine the effects of smoking during pregnancy on infant health.

00:14 And we use the birth weight as a proxy for the infant health.

00:19 And the main equation that we are interested in this one, birth weight is equal to beta 0 plus beta 1 times cigarettes plus beta 2 family income.

00:31 Here, birth weight is in ounces.

00:34 Cigarettes is the amount of cigarettes smoked per day during pregnancy, and the family income is in $1 ,000.

00:43 The question is, what is the most likely signed for beta 2? and we already had a hint for that in the question.

00:52 The last sentence in the question is, higher income generally results in access to better prenatal care as well as better nutrition for the mother.

01:02 So from that, we would expect beta 2 to be positive.

01:07 We would expect family income to have a positive effect on infant health, which is the birth weight here, keeping the amount of cigarettes, smokes, fixed.

01:20 And the second question is, what do we think about the correlation between cigarettes and family income is? do we expect them to be correlated? we would if we think that people have different consumption, different consumption behaviors in terms of smoking cigarettes when they have higher or lower incomes.

01:47 If we think that cigarettes are normal goods, then we would think that when people have higher incomes, they're going to smoke more.

01:55 Thus, we would expect a positive correlation.

01:58 But actually, cigarettes are like inferior goods.

02:01 So when family income goes up, people usually smoke less.

02:05 So we expect a negative correlation.

02:13 So let me also put this.

02:20 So we expect beta to be greater than zero, right? and correlation to be less than zero, negative correlation.

02:39 And the last question is now we're going to estimate this equation.

02:44 First using the family income then without the family income and we're going to compare our results okay so when we include the family income i get this estimates so i use data i just put progress be weight and cigarettes and family income and it gives me these estimates for constants my estimate is 116.

03:30 For the effect of cigarettes, 801 hats.

03:35 My estimate is negative .46.

03:43 And for the partial effect of family income, my estimate is .09.

03:53 The r squared from this regression is .0227.

04:05 And sample size is equal to 13808.

04:13 And when i run the regression without family income, i get the following estimates.

04:27 So just to differentiate, i'll just use a tilde here...

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