Question

The data set BWGHT contains data on births to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on \(n=1,388\) births: $$\widehat{b w g h t}=119.77-0.514 \mathrm{cigs}$$ i. What is the predicted birth weight when \(\operatorname{cigs}=0\) ? What about when \(\operatorname{cigs}=20\) (one pack per day)? Comment on the difference. ii. Does this simple regression necessarily capture a causal relationship between the child's birth weight and the mother's smoking habits? Explain. iii. To predict a birth weight of 125 ounces, what would cigs have to be? Comment. iv. The proportion of women in the sample who do not smoke while pregnant is about. \(85 .\) Does this help reconcile your finding from part (iii)?

Short answer

i. 119.77 oz for 0 cigs; 109.49 oz for 20 cigs; smoking lowers birth weight. ii. No, because of potential omitted variables. iii. Impossible to achieve with baseline model. iv. Yes, high non-smoking rate may skew results.

Step-by-step solution

Understand the Regression Equation

The given regression equation is \(\widehat{b w g h t} = 119.77 - 0.514 \times \text{cigs}\). This equation allows us to predict the birth weight based on the number of cigarettes smoked per day by the mother.

Predict Birth Weight for cigs = 0

Substitute \(\text{cigs} = 0\) into the regression equation: \(\widehat{b w g h t} = 119.77 - 0.514 \times 0 = 119.77\) ounces.

Predict Birth Weight for cigs = 20

Substitute \(\text{cigs} = 20\) into the regression equation: \(\widehat{b w g h t} = 119.77 - 0.514 \times 20 = 119.77 - 10.28 = 109.49\) ounces.

Comment on the Difference

The predicted birth weight decreases from 119.77 ounces to 109.49 ounces when the mother smokes one pack (20 cigarettes) per day. This suggests a negative relationship between smoking and birth weight.

Causal Interpretation

This simple regression does not necessarily capture a causal relationship because it might suffer from omitted variable bias. There could be other factors affecting birth weight that are not included in the model.

Solve for cigs to Get bwght = 125

To find the number of cigarettes (cigs) for a birth weight of 125 ounces: \(125 = 119.77 - 0.514 \times \text{cigs}\). Rearranging gives \(\text{cigs} = \frac{119.77 - 125}{-0.514} = -\frac{5.23}{0.514} \approx 10.17\).

Comment on the Solution for 125 ounces

The negative result indicates non-smoking or even cessation measures are unrealistic to achieve a birth weight of 125 ounces with smoking as the sole predictor, highlighting potential unreliability in prediction at certain points.

Reconcile Finding with Smoking Proportion

With about 85% of women in the sample not smoking during pregnancy, the high proportion of non-smokers may contribute to discrepancies in predicting birth weights, suggesting that the data may not adequately represent high-smoking populations and confounds predictions.

Key concepts

Regression Analysis

Regression analysis is a statistical method used to describe the relationship between a dependent variable and one or more independent variables. In our case, the regression equation is \[ \widehat{b w g h t} = 119.77 - 0.514 \times \text{cigs} \] This equation suggests how changes in the number of cigarettes smoked by the mother could affect the birth weight of the infant.
In economic and social sciences, regression analysis is widely used for predicting outcomes and uncovering relationships among variables.

• **Dependent Variable:** The outcome we are trying to predict, which is the birth weight (bwght) here.
• **Independent Variable:** This factor could potentially explain changes in the dependent variable. In this scenario, it is the number of cigarettes (cigs) smoked.

It helps us make sense of complex data by estimating how each additional unit of the independent variable is associated with the dependent variable, allowing predictions about outcomes under different scenarios.

Causal Inference

Causal inference seeks to establish whether a change in one variable truly causes a change in another.
In the problem, while the regression indicates a relationship between smoking and birth weight, it does not confirm a causal link due to potential confounding factors not considered in the model.
Factors such as maternal nutrition, prenatal care, and socioeconomic status can also influence birth weight.
Just because two variables show a relationship in a statistical model doesn't prove causation. It could mistakenly suggest causality where it doesn't exist.

• **Correlation vs. Causation:** While correlation is suggested by the regression results, causation is not guaranteed.
• **Omitted Variables:** These are variables not included in the analysis, which could affect the outcome, leading to potential causation errors.

To improve causal inference, additional methods like randomized controlled trials or using instrumental variables could be implemented, confirming that the relationship is indeed causal.

Omitted Variable Bias

Omitted variable bias occurs when a model leaves out one or more relevant variables, which could lead to incorrect conclusions about relationships between variables.
In the context of the birth weight example, ignoring variables such as the mother's age, income, and access to healthcare could skew the results significantly.
This bias means the coefficient of cigarettes in the model may not accurately reflect the true impact on birth weight.
Including more relevant variables in the regression could help address this issue, making the predictions more reliable and reducing the risk of drawing incorrect causal conclusions.

• **Impact on Results:** Could make the effect of smoking seem stronger or weaker than it actually is.
• **Solution:** Add more variables into the model to account for other influences on birth weight.

Ultimately, the trustworthiness of a regression analysis hinges on how well the model accounts for all potential influential factors.

Birth Weight Predictions

Predicting birth weight involves using the regression equation to forecast birth weight based on observable factors like smoking habits. The given regression predicts that more cigarettes correlate with lower birth weight.
For instance:

• **Non-Smoking Scenario:** Predicting for a non-smoker, we find the birth weight to be approximately 119.77 ounces.
• **Smoking Scenario:** For a mother smoking 20 cigarettes daily, the predicted birth weight drops to 109.49 ounces.

However, interpreting these predictions requires caution, especially when the model assumes smoking as the only factor affecting birth weight. Prediction limitations arise from the incorrect prediction of a birth weight of 125 ounces, calculated when negative cigarette consumption is required, which is unfeasible.
Improving accuracy necessitates incorporating a wider range of data points, encompassing various lifestyle and health factors that realistically affect birth weight.