1 Inferential Methods for Correlation and Regression Analysis


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1 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet ad the sample Regressio Lie were obtaied for describig ad measurig the quality ad stregth of the liear relatioship betwee two cotiuous variables. I this sectio a model for the uderlyig populatio will be developed. The this model will be used to obtai iferetial methods for this kid of data. 1.1 The Simple Liear Regressio Model Defiitio: The simple liear regressio model assumes that there is a lie with itercept α ad slope β, called the true populatio regressio lie, that describes the relatioship betwee the variable x ad y. Whe a value of the idepedet variable x is fixed ad a observatio o the depedet variable y ca be writte as, y = α + βx + e e is the error, ad we assume e N (0, σ). Basic Assumptios of the Simple Liear Regressio Model 1. The distributio of the radom deviatio e has populatio mea µ e =0. 2. The stadard deviatio of e is the same for ay particular value of x. It is deoted by σ. 3. The distributio of e is ormal. The model implies for the mea of the respose variable µ y = α + βx Radomess i e implies that the outcome of y for a give x is also radom, so that y is a radom variable. To summarize: That for every x, y is a ormal distributed radom variable with mea α + βx ad stadard deviatio σ. Iterpretatio: The slope β of the populatio regressio lie is the average chage i y associated with a 1uit icrease i x. The vertical itercept α is the mea of y whe x = 0. The value of σ describes the extet to which (x, y) observatios deviate vertically from the regressio lie. Diagram (compare Textbook pg.598) Diagram show small ad large σ. 1
2 1.2 Estimatig the Parameter of the Model The parameter of the model are α, β, ad σ. They ca be estimated (poit estimator ad cofidece iterval) by usig sample data (x 1, y 1 ), (x 2, y 2 ),..., (x, y ). Assume that the relatioship of x ad y ca be described by a Simple Liear Regressio Model ad that the observatios were draw idepedetly. Uder these assumptios poit estimates of α ad β are the itercept ad the slope of the least squares lie, respectively. That is poit estimate of β = b = S xy S xx poit estimate of α = a = ȳ b x where as before S xy = xy ( x)( y) ad S xx = x 2 ( x) 2. The estimated regressio lie is idetical to the least squares lie: ŷ = a + bx I additio to estimatig α ad β there is also a iterest i estimatig σ the spread of the radom deviatio of the regressio lie. This estimate will be based o the Sum of Squares for Error: SSE = (y ŷ) 2 where ŷ 1 = a + bx 1, ŷ 2 = a + bx 2,..., ŷ = a + bx are the fitted or predicted y values ad the residuals are y 1 ŷ 1, y 2 ŷ 2,..., y ŷ The residuals give the vertical distace of the observatios to the regressio lie. This makes SSE a measure of the extet to which the sample data spreads out about the estimated regressio lie. Estimatio of σ The statistic for estimatig the variace σ 2 is where The estimate of the stadard deviatio σ is s 2 = SSE 2 SSE = (y ŷ) 2 = y 2 a y b xy s = s 2 The Coefficiet of Determiatio ca ow be rewritte as r 2 = 1 SSE SS yy 2
3 where SS yy is SS yy = (y ȳ) 2 = y 2 ( y) 2 = S yy This equatio ow shows the iterpretatio of r 2 as the proportio of y variatio that ca be explaied by the model relatioship. σ measures the usual vertical deviatio of a poit (x, y) i the populatio from the populatio regressio lie. Similarly, s e measures the typical deviatio of a sample poit (x, y) of the sample regressio lie. 1.3 Iferece Cocerig the Slope of the Populatio Regressio Lie The slope B is the average chage i y, whe x is icreased by 1 uit. This iterpretatio makes B a iterestig parameter. I additio we fid, that if β = 0 the Populatio Regressio Lie would be parallel to the xaxis, thus a chage i x would t have a impact o y, which would mea that x ad y are idepedet. These two properties lead us to ask for a cofidece iterval for B, i order to be able to estimate B properly ad for a statistical test cocerig B. Before we ca obtai these methods we have to study the samplig distributio of the statistic to be used for estimatig B, which is b. Properties of the Samplig Distributio of b Whe the basic assumptios of the simple liear regressio model are met, the followig claims hold: 1. The populatio mea of b is β: µ b = β. Thus, b is a ubiased estimate of β. 2. The populatio stadard deviatio of b is σ b = σ Sxx 3. The statistic b has a ormal distributio (which is a cosequece of e beig ormally distributed). If σ b is large the estimate b does ot have to be close to the true β. I order for σ b to be small σ should be small, that meas little variatio about the populatio regressio lie, ad (or) S xx = (x x) 2 should be large, that meas far spread out x values are i favor for a smaller variatio i the estimate b for B. Stadardizig b Sice σ b depeds o the ukow σ it ca ot be used for a stadardizig b, i order to develop a statistic that ca be used for developig a cofidece iterval or test. 3
4 Istead use SE b = as estimate of σ b the stadard deviatio of b. This leads to: t = b β SE b which is t distributed with df = 2 (for a Simple Liear Regressio Model). This tstatistic does ot deped o ay ukow parameters beside β, so it ca be used for developig a cofidece iterval ad a test cocerig B. Cofidece Iterval for β A (1 α)% Cofidece Iterval for the slope β from a simple liear regressio model is give by: b ± (t critical value) 2 1 α SE b Where the t critical value is based o df = 2 ad C = 1 α (Table C). Example: Is cardiovascular fitess related to a athlete s performace i a 20km ski race? s S xx x= time to exhaustio ruig o a treadmill (i miutes) y= 20km ski time(i miutes) x y A scatterplot of this data shows a liear decrease i ski time by icreasig treadmill time. Straightforward calculatio gives: = 11 x = y = x 2 = xy = y 2 = 48,
5 From these calculate ad so that: S xy = xy ( x)( y) S xx = x 2 ( x) 2 = (106.3)(728.70) 11 = (106.3)2 11 = = b = S xy = = ad a = ȳ b x = = S xx The poit estimate for the decrease i ski time for every miute icrease i treadmill time is 2.33 miutes. Further more we get SSE = y 2 a y b xy = SS yy = y 2 ( y) 2 = r 2 = 1 SSE = SS yy s 2 = SSE 2 = s = s 2 = SE b = s Sxx = r 2 = is ot very high, but shows that about 63% of the variatio i skitime ca be explaied by the Liear Regressio Model. The estimate s = 2.19 shows that there is quite some variatio i the skitime for ay give treadmill time. The calculatio of a 95% cofidece iterval for β requires the t critical value with 2=9 degrees of freedom ad 1 α = 0.95: The resultig cofidece iterval is the b ± t critical value SE b = ± (2.26) (0.591) = ± = ( 3.671; 0.999) Based o the sample, we are 95% cofidet that the true average decrease i ski time associated with a oe miute icrease i treadmill time is betwee 1 ad 3.7 miutes. Hypothesis Test Cocerig β Hypotheses: Test type Upper tail H 0 : β 0 H a : β > 0 Lower tail H 0 : β 0 H a : β < 0 Two tail H 0 : β = 0 H a : β 0 5
6 Assumptios: Radom sample ad the Simple Liear Regressio Model applies to x ad y. Test statistic: Pvalue: t 0 = b SE b with df = 2 Test type Pvalue Upper tail P (t > t 0 ) Lower tail P (t < t 0 ) Two tail 2 P (t > abs(t 0 )) The two tail test of H 0 : β = 0 versus H a : β 0 is a test to assure that there is i fact a liear coectio betwee x ad y, it is also called Model Utility Test. Cotiue Example: Let s do the Model Utility Test for the example above at a sigificace level of α = Hypotheses: ad α = Do Model Utility Test. H 0 : β = 0 versus H a : β 0 2. Assumptios: Radom sample. Assume the relatioship of x ad y ca be described by a Simple Liear Regressio Model: 3. Test statistic: t 0 = b = = with df = 9 SE b Pvalue: This is a two tail test, thus pvalue= 2 P (t > abs(t 0 )) = = Decisio: Sice the pvalue α the ull hypothesis is rejected. 6. Coclusio: At sigificace level of 5% we coclude that β 0 ad there is a liear relatioship betwee ski time ad treadmill time. The Test Procedure: 1. Hypotheses: State the hypotheses ad the sigificace level, ad choose the appropriate test. 2. Assumptios: Check if the assumptios of that test are met. 3. Test statistic: Calculate the test statistic icludig the degrees of freedom if ecessary. 4. Pvalue: Obtai the pvalue, accordig to test type. 5. Decisio: Decide if the ull hypothesis ca be rejected, or ot. 6. Coclusio: Aswer the questio cocerig the problem stated for the situatio. 6
7 1.4 Checkig Model Adequacy Before relyig o the results based o a simple Liear Regressio Model, the model assumptios should be checked. These assumptios are mostly related to the distributio of the error e i the model equatio For the radom error e, we have to assume 1. For all x, e is ormally distributed, ad y = α + βx + e 2. The mea ad the stadard deviatio of e do ot deped o the value of x. If these assumptio are seriously violated the results based o the Simple Liear Regressio Model are ot reliable, ad may ot be true. This fact makes it ecessary to check for these assumptios. Residual Aalysis If the residuals e 1 to e from the populatio regressio lie α + βx would be available, we could use them to check for those assumptios. I ay applied situatio these values are ot kow, because the populatio regressio lie is ot kow. We have to rely o their estimates, the sample residuals (residuals from the least squares regressio lie, the estimated lie) y 1 ŷ 1 = y 1 (a + bx 1 ) y 2 ŷ 2 = y 2 (a + bx 2 )... y ŷ = y (a + bx ) Ay observatio that leads to very large residual should be examied for mistakes. How to determie if a residual is large? By ow we kow this is best be doe with by stadardizig the values ad tha judge them. How do we stadardize the residuals? We stadardize by subtractig the mea ad dividig by the stadard deviatio. But we kow that the populatio mea of the residuals is zero (µ e = 0). Ad it ca be determied that the stadard deviatio of the i th residual y i ŷ i is equal to ad the stadardized residual is equal to s 1 1 (x i x) 2 S xx. y i ŷ i s 1 1 (x i x) 2 S xx 7
8 Example 1 (Treadmill time ad ski time) For this data set the stadardized residuals are Treadmill Ski st. Res Is ay of these values ureasoably large? First do a histogram for the residuals to check if it is reasoable to assume they come from a ormal populatio. This histogram shows a uimodal ad symmetric distributio, without ay outliers. assumptio of ormality seems to be reasoable. The Plottig stadardized residuals versus tread mill time gives a visual presetatio of this data ad idicates if there is cocer that the stadard deviatio might vary for differet x. 8
9 This graph show that the stadardized residuals are evely spread above ad below the 0. The spread does ot chage with x. This plot does ot give ay idicatio that the Simple Liear Regressio Model assumptios are violated Applyig the Model Estimatio ad Predictio Oce we established that the model we foud describes the relatioship properly, we ca use the model for estimatio: estimate mea values of y, E(y) for give values of x. For example we might be iterested i the mea time a perso eed to fiish the ski race whe doig 10 miutes o the treadmill. predictio: ew idividual value based o the kowledge of x, For example, we wat to give the rage of ski times for people who ca do 10 miutes o the treadmill. The key to both applicatios of the model is the least squares lie, ŷ = a+bx, we foud before, which gives us poit estimators for both applicatios The ext step is to fid cofidece itervals for E(y) for a give x, ad for a idividual value y, for a give x. I order to do this we will have to discuss the distributio of ŷ for the two applicatios. Samplig distributio of ŷ = a + bx 1. The stadard deviatio of the distributio of the estimator ŷ of the mea value E(y) at a specific value of x, say x p 1 σŷ = σ + (x p x) 2 SS xx This is the stadard error of ŷ. 2. The stadard deviatio of the distributio of the predictio error, y ŷ, for the predictor ŷ of a idividual ew y at a specific value of x, say x p σ (y ŷ) = σ This is the stadard error of predictio (x p x) 2 SS xx
10 A (1 α)100% CI for E(y) at x = x p ŷ ± t s 1 + (x p x) 2 SS xx A (1 α)100% Predictio Iterval for y at x = x p ŷ ± t s (x p x) 2 SS xx Commet: Do a cofidece iterval, if iterested i the mea of y for a certai value of x, but if you wat to get a rage where a future value of y might fall for a certai value of x, the do a predictio iterval. Cotiue example: Accordig to the liear regressio model predict the middle 95% of ski times for people who do x p = 10 miutes o the treadmill? Fid a predictio iterval. For x p = 10 fid ŷ = (10) = With df = 9 t = ± 2.262(2.188) ( /11)2 +, gives ± which is [60.27, 70.65]. We expect that 95% of people who ca do 10 miutes o the treadmill to fiish the ski race i a time betwee ad miutes. Estimate the mea ski time for people who do x = 10 miutes o the treadmill at cofidece level of 95% ± 2.262(2.188) 1 11 ( /11)2 +, gives ± which is [63.90, 67.02]. We are 95% cofidet that the mea time to fiish the ski race for people who ca do 10 miutes o the treadmill falls betwee 63.9 ad 67 miutes. 10
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