MA 575, Linear Models : Homework 2

Transcription

1 MA 575, Linear Models : Homework 2 Question x i x 2 x 2 i 2x i x + x 2 x 2 i 2 x x i + n x 2 x 2 i 2 x x i + x x 2 i x i x x i xx i x i Question 2 RSSβ 0, β RSSβ 0, β β 0 RSSβ 0, β β y i β 0 + β x i 2 z 2 z zyi β 0+β x i 2y i β 0 + β x i 2 y i β 0 + β x i z 2 z zyi β 0+β x i 2y i β 0 + β x i x i 2 x i y i β 0 + β x i y i β 0 + β x i β 0 y i β 0 + β x i β

2 Question 3 2 RSSβ 0,β β 0 0 y i β 0 + β x i 0 nȳ nβ 0 nβ x 0 β 0 ȳ β x 2 RSSβ 0,β β 0 x i y i β 0 + β x i 0 x i y i nβ 0 x β n x i y i nȳ β x x β x 2 i 0 n x 2 i 0 Therefore, β x i y i nȳ x x 2 i n x 2 x i y i x i ȳ + y i x ȳ x x 2 i n x 2 x i xy i ȳ x 2 i x i x x i xy i ȳ x i x 2 Hence, β SXY Problem.2.2. When observing the graph about the dependence of soil temperature on month number, it does not seem that that the temperature is correlated to the month number..2.2 If we redraw the previous graph using the R-code below but make sure that the x-axis is at least for times longer than the y-axis, we obtain the graph below. We notice that a correlation now appears between the time of the year and the soil temperature. 2

3 Figure : Soil temperature vs the month number libraryalr3 # Attach the Mitchell data file attachmitchell # Plot the temperature on month number plotmitchell$month,mitchell$temp,xlab"month after January 976",ylab"Average Soil Temperature",asp/4 Problem Let s draw the scatter plot of the weight of our individuals versus their size. #Attach the htwt file attachhtwt #2.. Scatterplot Wt vs Ht plotht,wt,xlab"height",ylab"weight" grid Figure 2: Weight vs Height From this graph and because the sample s size is small n 0, it is hard to say if fitting a linear regression is appropriate. However, we can fit one and use statistical tests to answer that question To answer the first part of the question, we remind you that : x x i ; ȳ x i x 2 ; SY Y y i ȳ 2 ; SXY x i xy i ȳ To comupte these equations, we run the following code. y i y_bar meanwt x_bar meanht sumht-x_bar^2 SXY sumht-x_bar*wt-y_bar SYY sumwt-y_bar^2 data.framex_bar x_bar, y_bary_bar,, SXY SXY, SYY SYY x_bar y_bar SXY SYY

4 To obtain an estimation of the parameters of the linear regression, we use the formulas derived in Question 3. Beta SXY/ Beta0 y_bar - Beta*x_bar cbeta0,beta [] y_predicted Beta0 + Beta*Ht plotht,wt,xlab"height",ylab"weight" linesht,y_predicted grid Figure 3: Weight vs Height 2..3 Reminder : ˆσ 2 RSS, ˆβ ; n 2 V ar ˆβ 0 ˆσ 2 ; V ar ˆβ ˆσ2 ; cov, ˆβ ˆσ 2 x n lengthht RSS sumy_predicted - Wt^2 s2 RSS/n-2 var_beta0 s2*/ x_bar^2/ var_beta s2/ cov_beta0_beta -s2*x_bar/ cs2,var_beta0,var_beta,cov_beta0_beta [] We now have enough material to compute statistical tests and check wether or not a linear regression is appropriate for this sample. First, we will test the hypothesis H 0 : ˆβ0 0 at a significance level α 5%. We can formulate this problem as follows : By noticing that : We can write : P H0 rejecth 0 P H0 z α α P H0 z α z α α P H0 N n β 0, σ 2 + z α β 0 ˆβ0 β 0 n 4 + ẑ α β 0

5 Under the hypothesis H 0, we haveβ 0 0, therefore : P H0 z α ˆβ0 z α In our case, we cannot use he statistic proved that : because σ is an unknown parameter. However, it can be easily Therefore, the statistic : T ˆσ σ ˆσ 2 σ 2 χ2 n 2 n 2 V ar ˆβ τn 2 0 Therefore, we would reject the H 0 if : β 0 z α t α V ar where t α is the quantile defined such that : P t α T t α. or if: V ar ˆβ t α 0 The same logic works for β. We will reject the hypothesis β 0 if: where l α satisfies P l α U ˆβ l α. V ar ˆβ ˆβ l α V ar ˆβ U also follows a τn 2 distribution # t-test Statistics alpha 0.05 abscbeta0/sqrtvar_beta0,beta/sqrtvar_beta [] # t-test abscbeta0/sqrtvar_beta0,beta/sqrtvar_beta> qt-alpha/2,n-2 [] FALSE FALSE 5

6 In the case of β 0, to compute the two-tailed p-values we need to estimate : PT t T t which is equivalent to : p valβ0 2 P T t where t is the estimation of V ar By following the same train of thougts, one can obtain : p valβ 2 P U u where u is the estimation of ˆβ V ar ˆβ pval_beta0 2* - ptabsbeta0/sqrtvar_beta0,n-2 pval_beta 2* - ptabsbeta/sqrtvar_beta,n-2 cpval_beta0,pval_beta [] Source df SS MS F p-value Regression on Weight Residuals Table : ANOVA table Reminder : p val PU > F where U F, n 2. In addition, one can notice that F ˆβ V ar ˆβ Last but not least, the linear regression can be fitted and tested directly using the following R-code : HtWt.lm <- lmwt~ht, datahtwt summaryhtwt.lm anovahtwt.lm # Fit linear model # t-test # ANOVA 6