There are five questions as shown in the attached files. Someone please help me with the correct answers, thanks!
(a) Show that where the leftrhand side is called the odds ratio; Therefore1 log (Iii23$) 2 g + ,311'. (b) Show that the slope of the logistic regression curve pl::r} = 1H1 + s'w+1]] can be written as p(z)[l {{3}} Hint: differentiate 19(3) with respect to I." and then substitute for expressions that equal 13(3) and 1 {1.1:}. Question 1 [1 mark} Consider the ANUVA model with m. = 2 ciasses. Assume that all the standard as sumptions hold. and that the errors are Gaussian. {a} 1Write the model in standard {ANDVA} form and as a iinear regression model. (b) Derive the OLS estimator of the linear regression parameters. (-3] The F statistic for testing the signicance of the above regression is _ HEESSm 1} M Fatat m Fml,nm Show that m Regss = Eng-(E- W j=l and m 115 R53 2 2 [K1 Eda 1:1 i=1 (d) The usual t statistic for testing a difference between the means of two independent populations under equal variances is E ?g em=m= 2 where the standard error is _ _ f l ] SE[Y1 Y2} = S + '. 1'11 1'12 2 where the standard error is _ _ l ] SMYyY=S+q 1'11 1'12 with the pooled variance estimator 221(111 _ ?1}2 + Zity _ v2?! 1'11 + T12 2 32: Prove that in this case Farm 2 2 El'lali and therefore the two tests are equivalent; Question 2 [1 mark) Consider a cubic regression spline 1with one knot at E. The regression function has the form HI) = .59 + 1313 + .3232 + .5333 + 3.1{1' - E)?\" where U xESU [I_+= zg HEE> (a) Find a cubic polynomial 11(3): {11 + blz.' + 132 + d133, such that x) 2 fix) for all I .f E 0. Express :11, b1, c1 and d1 in terms of ,BD, ,81, 1821 .53! and J's-'1' (13) Find a cubic polynomial 13(1): (12 + bar + C2!2 + d213, such that x} = f2(I) for all m .f :=- [L Express :12, b2, (:2 and {la in terms of .30, .31, g, ,33, and 334. Together with {a}, this establishes that f[.1'} is a piecewise polynomial. (-3] Show that f1 [.5] = fgf}. That is, f[I} is continuous at E ((21) Show that fzf} = f}. That is, fit-Tl is oontinuous at E. (e) Show that f} = fE]. That is, f"[z) is continuous at {3. (f) Consider now the piecewise polynomial: 9(3) = 30 + #311 + 3232 + 3333 + aliw - 3+] + sHI - 311+ allw - Ell]: Show that 9(1) does not have continuous rst and seoond derimtives, and therefore this model is not a regression spline. QuEstion 3 [1 mark] In this question we stud}.r the interpretation of nonlinear models. For simplicity, dis regard the error let y represent the systematic part of the response variable. Suppose that y is a function of two predictors1 y = ffrl, $2}. a The metric effect of X1 is dened as the partial derivative ayf'arl. I The effect of a proportional change of $1 on y is dened as 11(6yfer1). I The instantaneous rate of return of y with respect to 1'1 is le'yfarl]. - The elasticity of X1 is dened as [ayfarlrlfyl Compute each of these four measures for the ve models below. Which measm'e yields the simplest result in each case? How can the several measures be interpreted? How would you t each model to the data. assuming convenient forms for the errors {eg1 additive errors for the rst three models)? (31.1; = .30 + .5131 + :3212 (Hy=&++me (c) y = ,5}. + .3111 + 3212 + 31112 (d) y = :5:me + 1311': + 13212} (E) y = nglzg\" 4 Question 4 [1 mark] The Bernoulli distribution has probability mass function fly] = er{1- lly where it is a parameter and y 6 {0,1}. (a) Write the loglikelihood for a sample 3;], gig, r r . , y\". (b) Derive the maximum likelihood estimate of fl\". ((3) Find an example of a business application in which we may want to model P(Y|X1, . . . , X\") as a Bernoulli distribution. (d) The Poisson distribution has probability mass mction M's"l where A is a parameter and y E N U D. We can show that the Poisson distribution has mean and variance A. 1 'Nrite the loglikelihood for a sample yl, 1.12,. . . , y\". (e) Derive the maximum likelihood estimate of A. (1") Find an example ofa business application in which we ma}.r 1want to model P(Y|X1, . . ., X\") as a Poisson distribution. Quastion 5 [1 mark] ln logistic regression {with one predictor in this ease}1 we use the logistic function Eiohm] 1 3:ch = = _, 1+ ems 1+ e was which implies that I]
