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Ευθαλία Διαμαντόπουλος
7 χρόνια πριν
Προβολές:

7 η

8 π 2 /3

14 χ 2

15 χ 2 t

17 k Y 0/0, 0/1,..., 3/3

18 π 1, π 2,..., π k k k 1 β ij Y I i = 1,..., I p (X i = x i1,..., x ip ) Y i J (j = 1,..., J) x i Y i = j π j (x i ) x i π j (x i ) x (n 1 (x),..., n J (x)) (π 1 (x),..., π j (x)) j n j (x) i Y i = j x i x logit

19 π odds = logit(π) = log = α + βx 1 π exp(α + βx) π(x) = 1 + exp(α + βx) β logit Y J logit(π j ) = log( π j π 1 ) = α j + β T j X j, j = 2,..., J β T j β j β j ( ) J 2 βj X j β T j ˆπ j = ˆπ 1 exp(β T j X j), j = 2,..., J. ˆπ 1 + ˆπ ˆπ j =1, 1 ˆπ 1 = 1 + J exp(β T j X j) π j = j=2 exp(x j β T j ), j = 2,..., J. 1 + J exp(β T j X j) J 1 j ˆπ j j=2

20 Y J ( ) J 2 J 1 π j (X) = Pr(J = j X) X j π j (X) = 1 J Y {π 1 (X),..., π j (X)} log π j(x) π J (X) = α j + β jx, j = 1,..., J 1 X J 1 J 1 log π α(x) π β (X) = log π α(x) π J (X) log π β(x) π J (X) {π j (X)} π j (X) = 1 + J 1 exp(α j + β T j X) h=1 exp(α h + β T h X) α J = 0 β J = 0 j = J α J = 0 β J = 0 j j j π j (x) = 1 J = 2 ( exp(α + βx) ) π(x) = 1 + exp(α + βx)

21 ξ ε i ξ = α + β 1 X i1 + + β k X ik + ε i ξ ξ m 1 m β α 1 < α 2 < < α m 1 Y 1 ξ α 1 2 α 1 < ξ α 2 Y = m 1 α m 2 < ξ α m 1 m α m 1 < ξ

22 Y Pr(Y i j) = Pr(ξ a j ) = Pr(α + β 1 X i1 + + β k X ik + ε i a j ) = Pr(ε i α j α β 1 X i1 β k X ik ) ε i logit [ Pr(Y i j) ] = log Pr(Y i) 1 Pr(Y i) = log Pr(Y i j) Pr(Y i > j) = α j α β 1 X i1 β k X ik β T x θ 3

23 θ j = α + b(j 1), j = 1,..., J 1 θ 1 = α Y 1, 2,..., k i y i X i p Y π π ij γ ij = Pr(Y i j) = p i1 + + p ij, j = 1,..., k Pr(X i < j X j ) Pr(Y j) odds = Pr(Y j) = Pr(Y j) 1 Pr(Y j) = p p j p j p k, j = 1,..., k 1

24 logit(γ ij ) = logit [Pr(Y j)] = log Pr(Y j), j = 1, 2,..., k 1. 1 Pr(Y j) Y j Y > j k logit(γ ij ) = logit [Pr(Y j)] = log π p j π j π k = α j + β T k X β k odds = Pr(Y y j X) = exp(α j + β T k X) 1 + exp(α j + β T, j = 1, 2,..., k k X) j α j α j j Pr(Y j) j X β k odds = Pr(Y y j X) = exp(α j + β T X) 1 + exp(α j + β T, j = 1, 2,..., k X) j = k 1 Pr(Y k) = 1 1 = 0

25 α j Y Y Y j α j α j j Pr(Y j) j X logit(γ ij ) = α j + β T X β T X

26 β T X α j β T X β T Y j X 1 X 2 odds = γ j (X 1 ) 1 γ j (X 1 ) γ j (X 2 ) 1 γ j (X 2 ) = exp(α j β T X 1 ) exp(α j β T X 2 ) = exp[βt (X 2 X 1 )] j X 1 X 2 X

27 X 2 X 1 = 1 exp( β ) Y j exp(β ) β T q p (q < p) q Y k x p Pr(Y y j ) = exp( α j β T X t T γ j ) 1 + exp( α j β T, j = 1, 2,..., k X t T γ j ) t q q γ j q t t T γ j j 1 j k γ 1 = 0

28 γ j = 0 j q t H 0 : γ j = 0, j (2 j k) γ 1 = 0 (α + β T x) Y y j = 1 y j > 1 (α + β T x) tγ j Pr(Y y j ) = exp( α j β T X t T γγ j ) 1 + exp( α j β T, j = 1, 2,..., k X t T γγ j ) Γ j Γ 1 = 0 γ q j c c 1 c 1

29 S F Y i j S i θ j 1 < S i < θ j θ 0 < θ 1 < < θ J 1 < θ J S J + 1 x i S i = α + β T x i + ε i, ε i N(0, σ 2 ) ε i α S i x i S i N(α + β T x i, σ 2 )

30 j ( γ ij = Pr(Y i j) = Pr(S i ) θ j = Pr Z i θ j α β T ) x i σ ( θj α β T ) x i = Φ σ Z i = S i α β T x i N(0, 1) Φ σ α σ γ ij = Φ(θ j β T x i ) ˆθ j = (θ j α) σ ˆβ = β σ. β

31 π 2 /3 π 2 /3

32 logit [ ] Pr(Y = j x) = log π j, j = 1,..., J 1. Pr(Y = j + 1 x) π j+1 log π j π J = log π j π j+1 + log π j π j log π j 1 π J log π j π j+1 = log π j π J log π j+1 π J, j = 1,..., J 1 ( ) J 2 log π j π j+1 = α j + β T X, j = 1,..., J 1 β J j log π J 1 j(x) π J (x) = α k + β T (J j)x, j = 1,..., J 1 k=j log π j(x) π J (x) = α j + β T u j, j = 1,..., J 1 Y

33 J 1 j = 1,, J 1 Pr(Y = j Y j) log π j π j π J, j = 1,..., J 1

34 π j+1 log, j = 1,..., J 1. π π j β j T j g(µ i ) = β T X i µ i y i = (y i1, y i2,..., y i,j 1 ) i g

35 x η η η x [ log ( )] µ 1 µ [Φ 1 (µ)] x β

36 η x β x k e β x k Pr(Y y 1 ) Pr(Y = y 1 ) x k e β

37 β n l(θ, β; y) = w i log π i i=1 i w i π i π ij I(Y i = j) I( ) ˆβ ˆθ β θ

38 α = [θ, β] H = 2 l(α; y) α α T α=ˆα I( ˆα) = H s. e( ˆα) = diag[ H]( ˆα) 1 ]

39 m 0 m 1 m 0 m 1 m 0 m 1 m 0 m 1 = 2(l 0 l 1 ) l 0 l 1 m 0 m 1 m 1 m 0 χ 2 m 0 m 1 m 0 m 1

40 D = D D χ 2 l = h w hj log ˆπ hj j ˆπ hj = w hj w h. w hj w h. h χ 2 χ 2

41 χ 2

42 β β j c 1

43 > summary( mentaldta) mental ses life Min. :1.000 Min. :0.00 Min. : st Qu.: st Qu.:0.00 1st Qu.:2.000 Median :2.000 Median :1.00 Median :4.000 Mean :2.325 Mean :0.55 Mean : rd Qu.: rd Qu.:1.00 3rd Qu.:6.250 Max. :4.000 Max. :1.00 Max. :9.000

45 > prop. vgam <- vglm( mental ~ ses + life, family = cumulative( parallel = T), data= mentaldta) > summary(prop.vgam) Call: vglm( formula = mental ~ ses + life, family = cumulative( parallel = T), data = mentaldta) Pearson residuals: Min 1Q Median 3Q Max logit(p[y <=1]) logit(p[y <=2]) logit(p[y <=3]) Coefficients: Estimate Std. Error z value Pr( > z ) ( Intercept): ( Intercept): ( Intercept): ** ses life ** --- Signif. codes: 0 *** ** 0.01 * Number of linear predictors: 3 35

46 Names of linear predictors: logit(p[y <=1]), logit(p[y <=2]), logit(p[y <=3]) Residual deviance: on 115 degrees of freedom Log - likelihood: on 115 degrees of freedom Number of iterations: 5 No Hauck - Donner effect found in any of the estimates Exponentiated coefficients: ses life logit [Pr(Y 1)] = X X 2 logit [Pr(Y 1)] = X X 2 logit [Pr(Y 3)] = X X 2 α i α 1 = , α 2 = , α 3 = ˆβ 1 = 0, 319 ˆβ 2 = 1.11 e 0,32 = 0, 72 X Y j j

47 e = 3 0 5% χ 0,05,2 = 0, 35 > prop. vgam0 <- vglm( mental ~1, family = cumulative( parallel = T), data= mentaldta) > summary(prop. vgam0) Call: vglm( formula = mental ~ 1, family = cumulative( parallel = T), data = mentaldta) Pearson residuals: Min 1Q Median 3Q Max logit(p[y <=1]) logit(p[y <=2]) logit(p[y <=3]) Coefficients: Estimate Std. Error z value Pr( > z ) ( Intercept): * ( Intercept): ( Intercept): ** --- Signif. codes: 0 *** ** 0.01 *

48 Number of linear predictors: 3 Names of linear predictors: logit(p[y <=1]), logit(p[y <=2]), logit(p[y <=3]) Residual deviance: on 117 degrees of freedom Log - likelihood: on 117 degrees of freedom Number of iterations: 1 No Hauck - Donner effect found in any of the estimates logit [Pr(Y 1)] = α 1 + β 11 X + β 21 X logit [Pr(Y 1)] = α 1 + β 12 X + β 22 X logit [Pr(Y 1)] = α 1 + β 11 X + β 21 X logit [Pr(Y 1)] = α 1 + β 13 X + β 23 X logit [Pr(Y 3)] = α 3 + β 11 X + β 21 X > cum. vgam <- vglm( mental ~ ses + life, family = cumulative, data= mentaldta) > summary(cum.vgam) Call: 38

49 vglm( formula = mental ~ ses + life, family = cumulative, data = mentaldta) Pearson residuals: Min 1Q Median 3Q Max logit(p[y <=1]) logit(p[y <=2]) logit(p[y <=3]) Coefficients: Estimate Std. Error z value Pr( > z ) ( Intercept): ( Intercept): ( Intercept): NA NA ses: ses: * ses: life: * life: * life: * --- Signif. codes: 0 *** ** 0.01 * Number of linear predictors: 3 Names of linear predictors: logit(p[y <=1]), logit(p[y <=2]), logit(p[y <=3]) Residual deviance: on 111 degrees of freedom Log - likelihood: on 111 degrees of freedom Number of iterations: 14 39

50 > deviance( prop. vgam)- deviance( cum. vgam) [1] > df. residual( prop. vgam)-df. residual( cum. vgam) [1] 4 > qchisq(0.05,4) [1] logit [Pr(Y 1)] = α 1 + β 11 X + β 2 X logit [Pr(Y 1)] = α 1 + β 12 X + β 2 X logit [Pr(Y 1)] = α 1 + β 11 X + β 2 X logit [Pr(Y 1)] = α 1 + β 13 X + β 2 X logit [Pr(Y 3)] = α 3 + β 11 X + β 2 X > summary(par.vgam) Call: vglm( formula = mental ~ ses + life, family = cumulative( parallel = F ~ ses), data = mentaldta) Pearson residuals: 40

51 Min 1Q Median 3Q Max logit(p[y <=1]) logit(p[y <=2]) logit(p[y <=3]) Coefficients: Estimate Std. Error z value Pr( > z ) ( Intercept): ( Intercept): ( Intercept): ** ses: ses: * ses: life ** --- Signif. codes: 0 *** ** 0.01 * Number of linear predictors: 3 Names of linear predictors: logit(p[y <=1]), logit(p[y <=2]), logit(p[y <=3]) Residual deviance: on 113 degrees of freedom Log - likelihood: on 113 degrees of freedom χ 2 > pchisq( deviance( prop. vgam)- deviance( par. vgam), + df. residual( prop. vgam)-df. residual( par. vgam), lower. tail = F) [1]

52 pchisq( deviance( par. vgam0)- deviance( par. vgam), + df. residual( par. vgam0)- df. residual( par. vgam), lower. tail = F) [1] J 1 j + 1, j = 1,..., J 1 log( π 1 π 2 ) = X X 2 log( π 2 π 3 ) = X X 2 log( π 3 π 4 ) = X X 2 e = > ad. vgam <- vglm( mental ~ ses + life, family =acat( reverse =T, parallel = T), data= mentaldta) 42

53 > summary( ad. vgam) Call: vglm( formula = mental ~ ses + life, family = acat( reverse = T, parallel = T), data = mentaldta) Pearson residuals: Min 1Q Median 3Q Max loge(p[y=1]/p[y=2]) loge(p[y=2]/p[y=3]) loge(p[y=3]/p[y=4]) Coefficients: Estimate Std. Error z value Pr( > z ) ( Intercept): ( Intercept): ( Intercept): ses life * --- Signif. codes: 0 *** ** 0.01 * Number of linear predictors: 3 Names of linear predictors: loge(p[y=1]/p[y=2]), loge(p[y=2]/p[y=3]), loge(p[y=3]/p[y=4]) Residual deviance: on 115 degrees of freedom Log - likelihood: on 115 degrees of freedom Number of iterations: 4

54 pchisq( deviance( ad. vgam)- deviance( prop. vgam), + df. residual( ad. vgam)- df. residual( prop. vgam), lower. tail = F) [1] 1 j = 1,, J1 π 1 log( ) = X X 2 π 2 + π 3 + π 4 π 2 log( ) = X X 2 π 3 + π 4 log( π 3 π 4 ) = X X 2 e = > con. vgam <- vglm( mental ~ ses + life, family =acat( reverse =F, parallel = T), data= mentaldta) > summary(con.vgam) Call: vglm( formula = mental ~ ses + life, family = acat( reverse = F, parallel = T), data = mentaldta) 44

55 Pearson residuals: Min 1Q Median 3Q Max loge(p[y=2]/p[y=1]) loge(p[y=3]/p[y=2]) loge(p[y=4]/p[y=3]) Coefficients: Estimate Std. Error z value Pr( > z ) ( Intercept): ( Intercept): ( Intercept): ses life * --- Signif. codes: 0 *** ** 0.01 * Number of linear predictors: 3 Names of linear predictors: loge(p[y=2]/p[y=1]), loge(p[y=3]/p[y=2]), loge(p[y=4]/p[y=3]) Residual deviance: on 115 degrees of freedom Log - likelihood: on 115 degrees of freedom > pchisq( deviance( con. vgam)- deviance( prop. vgam), +df. residual( con. vgam)- df. residual( prop. vgam), lower. tail = F) [1] 1

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