Least-squares Regression Line from Summary Data?

A study by the Berkeley Institute of Human Development (see the book Statistics by Freedman et al., listed in the back of the book) reported the following summary data for a sample of $n = 66$ California boys: $r = 0.79.$ At age 6, average height 47 in., standard deviation 1.7 in. At age 18, average height 68 in., standard deviation 2.6 in

Suppose that you wanted to predict the past value of 6-year-old height (Y) from knowledge of 18-year-old height (x). Find the equation for the appropriate least-squares line. What is the corresponding value of $S_e?$

$\hat Y$ = ? $S_e$ = ?

I'm absolutely lost here as my professor didn't go too in depth into tackling these word problems. Any help would be appreciated, thank you!

I'll leave the numerical computations to you. Here is an outline of what I think you need to do:

Data and notation. You have data $x_i$ at age 18: $n = 66, \bar x = 68, S_x = 2.6,$ and data $Y_i$ for the same $n$ boys at age 6: $\bar Y = 47, S_y = 1.7.$ Also, sample correlation $r_{x,Y} = 0.79.$

To establish notation: your regression model is $Y_i = \beta_0 + \beta_1 x_i + e_i,$ where $e_i$ are independent and from $\mathsf{Norm}(0, \sigma).$ The unknown parameters to be estimated are the y-intercept $\beta_0$ and the slope $\beta_1$ of the regression line, and the SD $\sigma$ of the 'noise' about the line.

Estimating parameters of the linear model. The estimate of $\beta_1$ is $b_1 = r_{x,Y}S_Y/S_x.$ Then because the regression line must pass through the 'center of gravity' $(\bar x, \bar Y)$ of the scatterplot of the $(x_, Y_i),$ you have $\bar Y = b_0 + b_1\bar x.$ You are given the information to find $b_1,$ and once you have that, you can solve to find $b_0.$ Then you can write the 'least-squares' (estimated) regression line as $$\hat Y_i = b_0 + b_1X_i.$$

Prediction. Now suppose we have a 'new' boy (whose measurements were not used to find the regression line). If his height at age 18 is $x_0,$ then his 'predicted' height at age 6 is $\hat Y_0 = b_0 + b_1x_0.$

Of course, we cannot expect this predicted height at 6 years will be exactly correct. First, the new boy will have his own discrepancy from the true regression line $Y = \beta_0 = \beta_1 x.$ Second, we do not know the true regression line, but only the 'least-squares' line with estimates $b_0$ and $b_1.$

If you look in your text or notes, you should find a formula to get a 95% prediction interval for $Y_0$ that gives an idea how far wrong your estimate of the 6-yr height might be. That formula requires you to know the estimate $S_e$ of the $\sigma$ in the original regression model. (Sometimes $S_e$ is written as $S_{Y|x}$ to emphasize we are using heights at 18 years to predict heights at 6 years.) One version of this formula is as follows:

$$\hat Y_0 \pm t^*S_e\sqrt{1 + \frac1n + \frac{(x_0 - \bar x)^2}{(n-1)S_x^2}},$$ where (at confidence level 95%) $t^*$ cuts 2.5% from the upper tail of Student's t distribution with $n - 2$ degrees of freedom. [Inside the square-root sign: the term $1$ recognizes that we are predicting for a 'new' value, the term $1/n$ recognizes that $b_0$ is only an estimate of $\beta_0$; and the third term recognizes that an error in estimating the slope becomes more serious the farther $x_0$ is from $\bar x.]$

I don't know what method you are supposed to use find $S_e.$ There are various possibilities. One formula that I have used is: $$S_e^2 = \frac{n-1}{n-2}S_Y^2(1 - r_{x,Y}^2).$$

Notes: (i) The formal definition of this estimate of $\sigma^2$ is $S_e^2 = \frac{\sum_i(\hat Y_i - Y_i)^2}{n-2} = \frac{1}{n-2}\sum_i d_i^2,$ where the $d_i$'s are called residuals. However, you have only summary information about the 'original' values $Y_i$ not the values themselves, so you can't use the definition (directly) to find $S_e.$

(ii) The estimated line is called the 'least-squares' line because $b_0$ and $b_1$ are chosen to minimize $\sum_i d_i^2.$

(iii) The displayed formula for $S_e^2$ has some intuitive value: If $r_{x,y}^2 = 1,$ then all points $(x_i, Y_i)$ lie exactly on the regression line and all $d_i \equiv 0;$ if $r_{x,Y}^2 \approx 0,$ then $S_e \approx S_Y,$ and the regression procedure has no predictive value.

(iv) Prediction of $\hat Y_0$ from a new $x_0$ is not recommended if $x_0$ is far outside the span $(\min(x_i), \max(x_i))$ of the $x_i$'s used to get the regression line.

(v) Ordinarily, one might try to use heights at 6 years to predict heights at 18 years. That would be an entirely different regression procedure. Correlation is symmetrical: $r_{x,y} = r_{y,x};$ but correlation is not: $S_{y|x}$ need not be the same as $S_{x|y}.$

• Thank you so much BruceET! This was very helpful. This helped a lot in understanding the textbook; I will reread it and compare it to your data notation (which helped immensely) and try to understand it more. Again, thanks for the explanation! – Sonja Dec 15 '17 at 4:40