The data set BWGHT contains data on births to women in the...

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)?

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)?

Key Concepts

Simple Linear Regression

Simple linear regression involves modeling the relationship between a dependent variable and a single independent variable using a straight line. The model is expressed through an intercept, which estimates the baseline level of the dependent variable when the independent variable is zero, and a slope, which indicates the average change in the dependent variable associated with a one-unit increase in the independent variable.

Interpretation of Regression Coefficients

The intercept in a regression model represents the predicted value of the outcome when the explanatory variable is set to zero, while the slope measures the effect of a one-unit change in the explanatory variable on the outcome. Understanding these coefficients is essential to interpret how changes in the independent variable are associated with variations in the dependent variable.

Prediction and Extrapolation

Using a regression equation to predict values involves substituting specific values of the independent variable into the fitted model. However, caution is needed if predictions require independent variable values that fall outside the observed data range (extrapolation), as the relationship estimated within the sample may not hold in such regions, leading to unrealistic or unreliable predictions.

Causal Inference in Observational Studies

In observational studies, regression analysis can identify associations between variables, but it does not by itself establish a causal link. Inferring causality requires careful consideration of potential confounding variables, omitted variable bias, and other factors that might influence both the independent and dependent variables, thus complicating the causal interpretation of the estimated relationship.

Transcript

00:01 Once again, welcome to a new problem.

00:04 This time we're dealing with regression elements, and there's always a powerful aspect to looking for relationships, relationships between quantitative variables.

00:26 And in this sense, you're saying you have your x, which is your independent, and then you have your y which is your dependent.

00:40 From a contextual aspect, your x variable is the explanatory variable, and your y variable is your response variable.

00:52 So you want to see the impact that the explanatory variable has on the response variable, and towards that you tend to build a model where we do have slope coefficients.

01:11 So this is a simple linear regression model where the e is our error and the b0 is our intercept.

01:24 So the point to which the value of x is zero.

01:28 So we don't have any influences.

01:29 We're going to make it zero.

01:31 And then of course, beta 1 is your slope coefficient.

01:38 And this is your slope coefficient that relates your x and your y value.

01:49 So in this particular problem, we're given a regression equation.

01:55 It's a simple linear regression equation based off of y hat equals to beta not plus beta 1x.

02:06 Y hat is the predicted y value because if you have a model with a bunch of data points, there's going to be a straight line which estimates the data points.

02:27 And since the straight line is based off of a sample, it's going to be a predicted equation.

02:38 So in this particular problem, we have an equation that relates the birth weight of children.

02:47 We have a hat on top of it because it's predicted.

02:50 And then the intercept is 119 .77 minus 0 .514x.

03:03 But in this case x is cigarettes so it's a relationship between bath weight and cigarettes so this one stands for the infant infant bath weight this is the infant bath weight value the y value is the infant bathroom, the response variable or the dependent variable, and then the independent variable cigs, that stands for the average number of cigarettes.

04:17 So this is the average number of cigarettes smoked per day during pregnancy.

04:32 So you know we want to see if there's a relationship between these two and we're given a regression model.

04:39 The first question is determine the birth weight when the mother smokes zero cigarettes.

05:00 During pregnancy.

05:07 And then the second thing is how about 20 cigarettes during pregnancy? we also want to check, compare the outcomes of these two inputs.

05:32 And part b was saying is the relationship, i was saying is the relationship between cigarettes and a bath weight causal.

06:06 So by causal, we mean that does smoking during pregnancy affect bath weight? does smoking during pregnancy affect bath weight? and then part c, given a birth weight of 1 .25.

06:41 Ounces, how many cigarettes would, how many cigarettes would cause this outcome? and that in part d, the proportion non -smokers in this sample, the proportion of non -smokers in this sample, the proportion of non -smokers and this sample is 0 .85.

07:37 How does this help the outcome in part c? so these are the questions presented and we're just going to jump in and figure out what the solutions are.

08:01 So if we have the number of cigarettes, if it's zero, then the equation becomes bath weight equals to 119 .7 minus 0 .514 and then you plug in 0 and obviously you can see your outcome is 119 .7 ounces.

08:24 And this happens because this part is going to cancel out.

08:30 And then in the second part, we're saying if the number of cigarettes is 20, we'll repeat the process with the bath weight.

08:39 But now you're plugging in 20 as your result right next to the slope coefficient and so you're going to end up with 109 .49.

08:57 We're going to end up with 109 .49.

09:03 So we just want to check that to make sure that we're getting the right answer.

09:09 Yes, so 109 .449.

09:12 So then that's the number of answers you're going to have.

09:19 That's part a.

09:23 And you can see that the relationship between smoking and bath weight is inverse, as mothers smoke more during pregnancy, their children's or infants, let's call it infants.

10:12 Not children.

10:14 Their influence, birth weight declines.

10:25 And then in the second part of the problem, we're looking to see if there's a causal relationship, we'll say mothers typically smoke before for the onset of pregnancy...

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