THE LEAST SQUARES REGRESSION LINE and R 2


 Shannon Valentine Phillips
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1 THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from the lie as small as possible. The book (which is ot writte for math majors) the just gives formulas for the coefficiets of this lie o p. 37. As math majors, you should kow why these are the right coefficiets, so here is a proof, with blaks for you to fill i. I ited to go through this i class, callig o studets to fill i the missig items A, B, Notatio: The least squares lie has equatio y = a + bx. The data poits are (x,y ),..., (x, y ). A. Draw a picture showig the lie, the i th data poit, y i ), ad the vertical distace d i from, y i ) to the lie y = ax + b. B. Express d i i terms of x i, y i, a, ad b: d i = C. Use your aswer to (B) to fid a expressio for the sum Q of the squares of the distaces of the data poits from the lie: Q = We wat to miimize Q. D. The idepedet variables i this expressio for Q are We eed the partials of Q with respect to these variables to be zero E. Why? So we have () 0 = Q/ a = Σ(y i  a  bx i ) () 0 = Q/ b = Σx i (y i  a  bx i ) Let s work first with (). Dividig by  ad distributig the sum gives (3) Σy i  Σa  Σbx i = 0. Now Σy i = y, Σa = a, ad Σbx i = bσx i = b( x ), so (3) simplifies to (4) y a  b x. Cacelig the s ad solvig for y gives
2 (5) y = a + b x. This tells us two thigs: a) The poit ( x, y ) lies o the least squares regressio lie. b) Oce we fid b, we ca fid a by a = y b x. Now we ll work with (). First divide by  to get (6) Σx i (y i  a  bx i ) = 0. Solve (5) for a ad substitute i (6): Regroup this to give Now solve for b: (7) Σx i (y i  y + b x  bx i ) = 0. (8) Σx i (y i  y )  b Σx i  x ) = 0. (9) b = [ Σx i (y i  y )]/[ Σx i  x )], which does t look like the formula o p. 37 for b. Let s look, however, at the book s formula: (0) b = r(s y /s x ). (formula from book) Rememberig the formula for r, amely, () r = " sy, (0) becomes () b = " s y " $ sy % ' & (from book formula)
3 3 = " (from book formula) Usig the defiitio of s x (ad cacelig  s) ow gives us (3) b = [ Σ  x )(y i  y )]/[ Σ  x ) ], (from book formula) which still is t the same as (9), but looks at least somewhat similar. We will show that it is i fact the same; statistics has lots of formulas that are somewhat like trig idetities, so your aswer may ot look like the aswer i the back of the book uless you do some algebra first. What we will use is the fact that the sum of the deviatios from the mea is zero. What this meas is that (4) Σ(y i  y ) = Σy i  Σ y = y  y (F. Why? ) = 0. Usig (4) ca make the umerator of (3) look like the umerator of (9): (5) Σ  x )(y i  y ) = Σx i (y i  y )  Σ x (y i  y ) = Σx i (y i  y )  x Σ(y i  y ) = Σx i (y i  y ) by (4) Similarly, usig the x versio of (4) we ca make the deomiator of (4) look like the deomiator of (9). So the two formulas, (9) ad (0), really say the same thig. Commet: The origial equatios () ad () (rather, the equatios oce we divide by ) have further uses. I the termiology of Sectio.4, y i  y ˆ i (= y i  a  bx i ) is called a residual. Sice it is the residual correspodig to the i th data poit, y i ), we will call it the i th residual ad deote it e i. Thus: e i = y i  ˆ y i = y i  a  bx i. Equatio () cleaed up the says that (6) Σe i = 0. That is, the sum of the residuals is zero. Equatio () i cleaed up form (i.e., equatio (6)) says (7) Σx i e i = 0.
4 4 (This ca be thought of as sayig that the sum of the residuals weighted by the x observatios is zero.) Usig these, we also have (8) Σ ˆ y i e i = Σ(a + bx i )e i = aσe i + b Σx i e i = 0 (by (6) ad (7)) (Thus the sum of the residuals weighted by the predicted values is zero.) II. The book also makes a assertio about the coectio of r with regressio that we will ow prove. First, we eed to fid ˆ y, the mea of the predicted values. I fact, (9) ˆ y = (/)Σ ˆ y i = (/)Σ (a + bx i ) = (/)Σ a + (/)(bσx i ) = a + b x = y by (5). I other words, the y observatios ad their predicted values have the same mea! Now use the formulas to reexpress the least squares lie: ˆ y = a + bx = ( y b) + bx = y + b(x  x ), so ˆ y  y = b(x  x ) = r(s y /s x )(x  x ) Applyig this to the data poits, we have i particular that for each i, ˆ y i  y = r(s y /s x )  x ).
5 5 Summig over i, dividig by , ad usig (9), we get var( ˆ y ) = ( y ˆ i = " s " r y s x ( x i " x ) = r s y s var(x) = r (s y ) = r var(y) x So (0) r = var( ˆ y )/var(y). Sice r is at most, we see i particular that the variace of y ˆ is at most the variace of y. We ca thik of y ˆ as the part of y that ca be explaied by regressio of y o x. Hece we ca say that r is the fractio of the variace of y that is explaied by regressio of y o x. (This is ot quite the same as the assertio o p. 4, but the two assertios are equivalet, sice the variatio i y refers to ()var(y).) Commet: With a little more work it is possible to show somethig eve stroger, amely: () var(y) = var( ˆ y ) + var(e). I words: The variace of the observatios is the sum of the variace of the predicted values ad the variace of the residuals. Whe this is writte out as summatios (ad multiplied by  to get less messy expressios), it looks like a typographical error or arithmetic mistake: () Σ[( y ˆ i  y ) + e i ] = Σ ( y ˆ i  y ) + Σe i However, if you look at it aother way, you might see that it is somethig like a Pythagorea Theorem. Equatio () says that there are two cotributios to var(y): the cotributio var( y ˆ ) from the predicted values  that is, from the regressio alog x  ad the cotributio var(e) from the residuals. (This idea of cotributios to variace from differet sources occurs i some advaced topics i statistics, such as Aalysis of Variace ad Multivariate Aalysis.)
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