Responsi 7 STA543 Analisis Data Kategorik

Model Regresi Logit Multinomial

setwd("D:\\Kuliah S2 IPB\\Bahan Kuliah\\Semester 2 SSD 2020\\STA543 ADK\\Responsi\\R\\UTS\\")

Pendahuluan

Regresi logistik biner digunakan untuk memodelkan hubungan antara peubah respon yang terdiri dari dua kategori dengan satu atau lebih peubah penjelas. Peubah penjelasnya bisa berupa data kontinu atau kategorik. Sedangkan regresi logistik multinomial digunakan untuk memodelkan hubungan antara peubah respon lebih dari dua kategori dengan satu atau lebih peubah penjelas. Peubah penjelasnya bisa berupa data kontinu atau kategorik. Peubah responnya berskala nominal (tidak ada tingkatan).

Soal 1

Input Data

##=====================##
# INPUT DATA
##=====================##
size <- factor(rep(c("<=2.3",">2.3"),40))
food <-factor(rep(c("Fish","Inver","Rept","Bird","Other"),rep(16,5)))
gender<-rep(factor(rep(c("Male","Female"),rep(2,2))),20)
lake<-rep(factor(rep(c("Han","Okl","Tra","Geo"),rep(4,4))),5)
counts<-c(7,4,16,3,2,13,3,0,3,8,2,0,13,9,3,8,
1,0,3,0,2,7,9,1,7,6,4,1,10,0,9,1,
0,0,2,1,0,6,1,0,1,6,1,0,0,0,1,0,
0,1,2,2,0,0,0,1,0,3,1,0,2,1,0,0,
5,2,3,3,1,0,2,0,1,5,4,0,2,2,1,1)

Penentuan Referensi

# Referensi
size <- relevel(size, ref=">2.3")
food <- relevel(food, ref="Fish")
gender <- relevel(gender, ref="Female")
lake <- relevel(lake, ref="Geo")

Membuat Dataframe

# Membuat data frame
makan <- data.frame(lake, gender, size, food, counts)
datafood <- makan[rep(row.names(makan),counts),1:4]

# Cek Struktur dan ukuran data
View(datafood)
dim(datafood)

## [1] 219   4

head(datafood)

##     lake gender  size food
## 1    Han   Male <=2.3 Fish
## 1.1  Han   Male <=2.3 Fish
## 1.2  Han   Male <=2.3 Fish
## 1.3  Han   Male <=2.3 Fish
## 1.4  Han   Male <=2.3 Fish
## 1.5  Han   Male <=2.3 Fish

A. Lakukan pemodelan regresi logistik multinomial pada data tersebut dengan peubah responnya adalah tipe makanan utama alligator dan peubah bebasnya adalah Lake (L) dan Size (S). Bandingkan hasilnya dengan buku Agresti serta berikan interpretasi pada tiap nilai dugaan parameter model.

##=====================## 
# MODEL REGRESI LOGISTIK MULTINOMIAL DENGAN PEUBAH BEBAS LAKE+SIZE
##=====================## 
library("foreign")
library("nnet")
model_1 <- multinom(food ~ lake + size, data=datafood)

## # weights:  30 (20 variable)
## initial  value 352.466903 
## iter  10 value 271.607785
## iter  20 value 270.046051
## final  value 270.040140 
## converged

summary(model_1)

## Call:
## multinom(formula = food ~ lake + size, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan      lakeOkl  lakeTra  size<=2.3
## Bird    -2.093358  0.6954256 -0.652622721 1.088098 -0.6306329
## Inver   -1.549021 -1.6581178  0.937237973 1.122002  1.4581457
## Other   -1.904343  0.8263115  0.005792737 1.516461  0.3315514
## Rept    -3.314512  1.2428408  2.458913302 2.935262 -0.3512702
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3
## Bird    0.6622972 0.7813123 1.2020025 0.8417085 0.6424863
## Inver   0.4249185 0.6128466 0.4719035 0.4905122 0.3959418
## Other   0.5258313 0.5575446 0.7765655 0.6214371 0.4482504
## Rept    1.0530577 1.1854031 1.1181000 1.1163844 0.5800207
## 
## Residual Deviance: 540.0803 
## AIC: 580.0803

A. MEMBANDINGKAN OUTPUT R DENGAN OUTPUT SAS

Output SAS:

Berdasarkan kedua gambar di atas, dapat dilihat bahwa pemodelan regresi logistik multinomial baik menggunakan Program R maupunSAS memberikan hasil analisis yang sama.

A. INTERPRETASI KOEFISIEN REGRESI

# odds = exponensial parameter model
exp(summary(model_1)$coefficients)

##       (Intercept)   lakeHan    lakeOkl   lakeTra size<=2.3
## Bird   0.12327247 2.0045620  0.5206784  2.968624 0.5322548
## Inver  0.21245592 0.1904972  2.5529204  3.070995 4.2979825
## Other  0.14892051 2.2848753  1.0058095  4.556075 1.3931278
## Rept   0.03635178 3.4654441 11.6920989 18.826431 0.7037936

Persamaan prediksi odds dimana alligator memilih memakan bird dibandingkan dengan fish

Persamaan prediksi odds dimana alligator memilih memakan bird dibandingkan dengan fish adalah:

Interpretasi: exp(−2.09)= 0.123.

Untuk alligator berukuran > 2.3 m di Lake George, odds alligator memakan bird dibandingkan alligator memakan fish adalah 0.123.

exp(−0.63)= 0.532.

Untuk alligator berukuran ≤ 2.3 m, odds alligator memakan bird dibandingkan fish adalah 0.532 kalinya odds alligator beukuran > 2.3 m.

exp(0.7)= 2.005.

Untuk alligator di Lake Hancock, odds alligator memakan bird dibandingkan fish adalah 2.005 kalinya odds alligator di Lake George.

exp(−0.65)= 0.521.

Untuk alligator di Lake Oklawaha, odds alligator memakan bird dibandingkan fish adalah 0.521 kalinya odds alligator di Lake George.

exp(1.09)= 2.97.

Untuk alligator di Lake Trafford, odds alligator memakan bird dibandingkan fish adalah 2.97 kalinya odds alligator di Lake George.

Persamaan prediksi odds dimana alligator memilih memakan invertebrata dibandingkan dengan fish

Persamaan prediksi odds dimana alligator memilih memakan invertebrata dibandingkan dengan fish adalah:

Interpretasi: exp(-1.55)= 0.21.

Untuk alligator berukuran > 2.3 m di Lake George, odds alligator memakan invertebrata dibandingkan fish adalah 0.21.

exp(1.46)= 4.3.

Untuk alligator berukuran ≤ 2.3 m, odds alligator memakan invertebrata dibandingkan fish adalah 4.3 kalinya odds alligator beukuran > 2.3 m.

exp(−1.66)= 0.19.

Untuk alligator di Lake Hancock, odds alligator memakan invertebrata dibandingkan fish adalah 0.19 kalinya odds alligator di Lake George.

exp(0.94)= 2.55.

Untuk alligator di Lake Oklawaha, odds alligator memakan invertebrata dibandingkan fish adalah 2.55 kalinya odds alligator di Lake George.

exp(1.12)= 3.07.

Untuk alligator di Lake Trafford, odds alligator memakan invertebrata dibandingkan fish adalah 3.07 kalinya odds alligator di Lake George.

Persamaan prediksi odds dimana alligator memilih memakan others dibandingkan dengan fish

Persamaan prediksi odds dimana alligator memilih memakan others dibandingkan dengan fish adalah:

Interpretasi: exp(−1.9)= 0.15.

Untuk alligator berukuran > 2.3 m di Lake George, odds alligator memakan others dibandingkan fish adalah 0.15.

exp(0.33)= 1.39.

Untuk alligator berukuran ≤ 2.3 m, odds alligator memakan others dibandingkan fish adalah 1.39 kalinya odds alligator beukuran > 2.3 m.

exp(0.83)= 2.28.

Untuk alligator di Lake Hancock, odds alligator memakan others dibandingkan fish adalah 2.28 kalinya odds alligator di Lake George.

exp(0.01)= 1.01.

Untuk alligator di Lake Oklawaha, odds alligator memakan others dibandingkan fish adalah 1.01 kalinya odds alligator di Lake George.

exp(1.52)= 4.56.

Untuk alligator di Lake Trafford, odds alligator memakan others dibandingkan fish adalah 4.56 kalinya odds alligator di Lake George.

Persamaan prediksi odds dimana alligator memilih memakan reptile dibandingkan dengan fish

maan prediksi odds dimana alligator memilih memakan reptile dibandingkan dengan fish adalah:

Interpretasi:

exp(−3.31)= 0.036.

Untuk alligator berukuran > 2.3 m di Lake George, odds alligator memakan reptile dibandingkan fish adalah 0.036.

exp(−0.35)= 0.7.

Untuk alligator berukuran ≤ 2.3 m, odds alligator memakan reptile dibandingkan fish adalah 0.7 kalinya odds alligator beukuran > 2.3 m.

exp(1.24)= 3.47.

Untuk alligator di Lake Hancock, odds alligator memakan reptile dibandingkan fish adalah 3.47 kalinya odds alligator di Lake George.

exp(2.46)= 11.7.

Untuk alligator di Lake Oklawaha, odds alligator memakan reptile dibandingkan fish adalah 11.7 kalinya odds alligator di Lake George.

exp(2.94)= 18.83.

Untuk alligator di Lake Trafford, odds alligator memakan reptile dibandingkan fish adalah 18.83 kalinya odds alligator di Lake George.

B. Lakukan pemodelan seperti pada poin (a) di atas, tetapi peubah bebasnya adalah Lake (L), Size (S), dan Gender (G). Peubah mana saja (L, S, G) yang berpengaruh nyata? Gunakan uji Deviance untuk 𝛼 = 0.05.

##=====================## 
# MODEL REGRESI LOGISTIK MULTINOMIAL DENGAN PEUBAH BEBAS LAKE+SIZE+GENDER
##=====================##
model_2 <- multinom(food ~ lake + size + gender, data=datafood)

## # weights:  35 (24 variable)
## initial  value 352.466903 
## iter  10 value 270.967533
## iter  20 value 268.934907
## final  value 268.932740 
## converged

summary(model_2)

## Call:
## multinom(formula = food ~ lake + size + gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan     lakeOkl  lakeTra  size<=2.3 genderMale
## Bird    -1.701750  0.5751767 -0.55073594 1.236877 -0.7302898 -0.6064571
## Inver   -1.167210 -1.7805263  0.91314471 1.155786  1.3362563 -0.4629756
## Other   -1.721273  0.7665622  0.02600333 1.557716  0.2905663 -0.2525879
## Rept    -2.858940  1.1294537  2.53024455 3.061047 -0.5570885 -0.6276206
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    0.7690488 0.7952037 1.2099728 0.8660879 0.6522819  0.6888490
## Inver   0.5337511 0.6232106 0.4761165 0.4927850 0.4111930  0.3955225
## Other   0.6313811 0.5685499 0.7777727 0.6256727 0.4599253  0.4663465
## Rept    1.1456331 1.1928006 1.1221172 1.1297303 0.6466081  0.6852751
## 
## Residual Deviance: 537.8655 
## AIC: 585.8655

# derajat bebas
abs(model_1$edf-model_2$edf)

## [1] 4

# Mencari Chi Square tabel df=4 alpa=5%
qchisq(0.05, 4, lower.tail=FALSE)

## [1] 9.487729

##=====================## 
# MODEL REGRESI LOGISTIK MULTINOMIAL DENGAN PEUBAH BEBAS LAKE+GENDER
##=====================##
model_3 <- multinom(food ~ lake + gender, data=datafood)

## # weights:  30 (20 variable)
## initial  value 352.466903 
## iter  10 value 279.421076
## iter  20 value 277.733691
## final  value 277.732884 
## converged

summary(model_3)

## Call:
## multinom(formula = food ~ lake + gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan     lakeOkl   lakeTra genderMale
## Bird  -2.12320607  0.4818741 -0.47393004 1.2811363 -0.4245404
## Inver  0.01048504 -1.7605713  0.59314476 0.9609816 -0.8577619
## Other -1.51431307  0.7842964 -0.07500933 1.4829188 -0.2871253
## Rept  -3.37256725  1.1400408  2.55949903 3.0361432 -0.1834060
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra genderMale
## Bird    0.7212115 0.7960695 1.1922056 0.8381776  0.6476502
## Inver   0.3584792 0.6187324 0.4425069 0.4722548  0.3694164
## Other   0.5318941 0.5701319 0.7661014 0.6158038  0.4542938
## Rept    1.0863463 1.1945082 1.1088625 1.1134113  0.5873043
## 
## Residual Deviance: 555.4658 
## AIC: 595.4658

# derajat bebas
abs(model_3$edf-model_2$edf)

## [1] 4

# Mencari Chi Square tabel df=4 alpa=5%
qchisq(0.05, 4, lower.tail=FALSE)

## [1] 9.487729

##=====================## 
# MODEL REGRESI LOGISTIK MULTINOMIAL DENGAN PEUBAH BEBAS LAKE+GENDER
##=====================##
model_4 <- multinom(food ~ size + gender, data=datafood)

## # weights:  20 (12 variable)
## initial  value 352.466903 
## iter  10 value 294.669539
## final  value 294.091727 
## converged

summary(model_4)

## Call:
## multinom(formula = food ~ size + gender, data = datafood)
## 
## Coefficients:
##       (Intercept)  size<=2.3  genderMale
## Bird   -1.2790620 -0.7533726 -0.61466393
## Inver  -0.9638861  0.9210878 -0.09004848
## Other  -1.0905582  0.2328891 -0.19634008
## Rept   -1.2170166 -0.8681379 -0.03140815
## 
## Std. Errors:
##       (Intercept) size<=2.3 genderMale
## Bird    0.5806349 0.6439387  0.6337949
## Inver   0.4001729 0.3730328  0.3538182
## Other   0.4584432 0.4373966  0.4378935
## Rept    0.5455407 0.5629537  0.5693974
## 
## Residual Deviance: 588.1835 
## AIC: 612.1835

# derajat bebas
abs(model_4$edf-model_2$edf)

## [1] 12

# Mencari Chi Square tabel df=12 alpa=5%
qchisq(0.05, 12, lower.tail=FALSE)

## [1] 21.02607

Berdasarkan hasil pengujian di atas dapat disimpulkan bahwa peubah yang berpengaruh nyata terhadap tipe makanan utama Alligator adalah peubah Lake dan peubah Size.

C. Diketahui seekor alligator berasal dari danau Trafford, berukuran pendek dan berkelamin jantan

##=====================##
# POIN C
##=====================##
#predict reglog
new=data.frame(size=as.factor("<=2.3"),gender=as.factor("Male"),lake=as.factor("Tra"))
#model2
predict(model_2,newdata=new,"probs")

##       Fish       Bird      Inver      Other       Rept 
## 0.20881597 0.03446118 0.49438276 0.18417282 0.07816727

Berdasarkan output dari program R peluang yang paling besar adalah aliigator tersebut akan memakan Invertebrata.

D. Tentukan model terbaik dengan peubah bebasnya adalah Lake (L), Size (S), dan Gender (G) serta semua interaksinya (LS, LG, dan LSG). Gunakan uji Deviance untuk 𝛼 = 0.05.

##=====================##
# POIN D
##=====================##
# model Y~X1
model_1c1 <- multinom(food ~ lake,data=datafood)

## # weights:  25 (16 variable)
## initial  value 352.466903 
## iter  10 value 281.030560
## iter  20 value 280.583926
## final  value 280.583844 
## converged

summary(model_1c1)

## Call:
## multinom(formula = food ~ lake, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan     lakeOkl   lakeTra
## Bird   -2.3982809  0.6065505 -0.49188808 1.2197770
## Inver  -0.5008393 -1.5137909  0.55488981 0.8263598
## Other  -1.7048477  0.8686390 -0.08689071 1.4425750
## Rept   -3.4962205  1.1937161  2.55175319 3.0107928
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra
## Bird    0.6031161 0.7727128 1.1912598 0.8310587
## Inver   0.2833774 0.6029744 0.4341550 0.4612870
## Other   0.4438217 0.5542879 0.7654142 0.6114775
## Rept    1.0148749 1.1817886 1.1083250 1.1099095
## 
## Residual Deviance: 561.1677 
## AIC: 593.1677

# model Y~X2
model_1c2 <- multinom(food ~ size,data=datafood)

## # weights:  15 (8 variable)
## initial  value 352.466903 
## iter  10 value 294.670879
## final  value 294.606678 
## converged

summary(model_1c2)

## Call:
## multinom(formula = food ~ size, data = datafood)
## 
## Coefficients:
##       (Intercept)  size<=2.3
## Bird    -1.727214 -0.5551882
## Inver   -1.034070  0.9489120
## Other   -1.241709  0.2943162
## Rept    -1.241705 -0.8583649
## 
## Std. Errors:
##       (Intercept) size<=2.3
## Bird    0.3836949 0.6063277
## Inver   0.2910708 0.3568648
## Other   0.3148735 0.4149523
## Rept    0.3148729 0.5349960
## 
## Residual Deviance: 589.2134 
## AIC: 605.2134

# model Y~X3
model_1c3 <- multinom(food ~ gender,data=datafood)

## # weights:  15 (8 variable)
## initial  value 352.466903 
## iter  10 value 301.192714
## final  value 301.129428 
## converged

summary(model_1c3)

## Call:
## multinom(formula = food ~ gender, data = datafood)
## 
## Coefficients:
##       (Intercept) genderMale
## Bird   -1.7635851 -0.3680370
## Inver  -0.2231478 -0.3578812
## Other  -0.9162928 -0.2708745
## Rept   -1.7635569  0.2509570
## 
## Std. Errors:
##       (Intercept) genderMale
## Bird    0.4418570  0.5958549
## Inver   0.2535467  0.3339731
## Other   0.3162281  0.4153372
## Rept    0.4418517  0.5376858
## 
## Residual Deviance: 602.2589 
## AIC: 618.2589

# model Y~X1+X2+X3+X1X2
model_1c4 <- multinom(food ~ lake +size + gender + lake*size, 
data=datafood)

## # weights:  50 (36 variable)
## initial  value 352.466903 
## iter  10 value 265.274636
## iter  20 value 260.069949
## iter  30 value 258.937775
## iter  40 value 258.824166
## iter  50 value 258.819006
## final  value 258.818962 
## converged

summary(model_1c4)

## Warning in sqrt(diag(vc)): NaNs produced

## Call:
## multinom(formula = food ~ lake + size + gender + lake * size, 
##     data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan     lakeOkl   lakeTra   size<=2.3  genderMale
## Bird    -2.442347   1.928711   0.6706186  2.282905  0.92900200 -0.84648228
## Inver   -2.453306 -12.256978   2.7331906  3.111918  3.17363446 -0.81592590
## Other   -1.688556   1.392344 -26.0432791  1.299438  0.07643898 -0.08264057
## Rept   -28.472136  26.922121  28.6961637 29.221745 26.38883928 -1.07334267
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3
## Bird         -2.6180164        -37.538905        -1.9359701
## Inver        10.0209021         -2.448568        -2.6137822
## Other        -0.8062165         27.172630         0.3637365
## Rept        -26.9993336        -27.961392       -27.5229971
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    1.0695771 1.2468205 1.5193189 1.3106646 1.2892933  0.7430086
## Inver   1.0470055 0.3370560 1.1530087 1.1841238 1.0964497  0.4448919
## Other   0.6785145 0.8580021 0.4916661 0.8711876 0.8925871  0.4749617
## Rept    0.8453331 1.0511858 0.7479391 0.7710755 0.6558217  0.7692781
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3
## Bird          1.6535383               NaN          1.897784
## Inver         0.3370534         1.3660324          1.361887
## Other         1.1500995         0.4916661          1.258627
## Rept          1.1866462         1.2780398          1.085509
## 
## Residual Deviance: 517.6379 
## AIC: 589.6379

# model Y~X1+X2+X3+X1X2
model_1c4 <- multinom(food ~ lake +size + gender + lake*size, 
data=datafood)

## # weights:  50 (36 variable)
## initial  value 352.466903 
## iter  10 value 265.274636
## iter  20 value 260.069949
## iter  30 value 258.937775
## iter  40 value 258.824166
## iter  50 value 258.819006
## final  value 258.818962 
## converged

summary(model_1c4)

## Warning in sqrt(diag(vc)): NaNs produced

## Call:
## multinom(formula = food ~ lake + size + gender + lake * size, 
##     data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan     lakeOkl   lakeTra   size<=2.3  genderMale
## Bird    -2.442347   1.928711   0.6706186  2.282905  0.92900200 -0.84648228
## Inver   -2.453306 -12.256978   2.7331906  3.111918  3.17363446 -0.81592590
## Other   -1.688556   1.392344 -26.0432791  1.299438  0.07643898 -0.08264057
## Rept   -28.472136  26.922121  28.6961637 29.221745 26.38883928 -1.07334267
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3
## Bird         -2.6180164        -37.538905        -1.9359701
## Inver        10.0209021         -2.448568        -2.6137822
## Other        -0.8062165         27.172630         0.3637365
## Rept        -26.9993336        -27.961392       -27.5229971
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    1.0695771 1.2468205 1.5193189 1.3106646 1.2892933  0.7430086
## Inver   1.0470055 0.3370560 1.1530087 1.1841238 1.0964497  0.4448919
## Other   0.6785145 0.8580021 0.4916661 0.8711876 0.8925871  0.4749617
## Rept    0.8453331 1.0511858 0.7479391 0.7710755 0.6558217  0.7692781
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3
## Bird          1.6535383               NaN          1.897784
## Inver         0.3370534         1.3660324          1.361887
## Other         1.1500995         0.4916661          1.258627
## Rept          1.1866462         1.2780398          1.085509
## 
## Residual Deviance: 517.6379 
## AIC: 589.6379

# model Y~X1+X2+X3+X1X3
model_1c5 <- multinom(food ~ lake +size + gender +lake*gender,
data=datafood)

## # weights:  50 (36 variable)
## initial  value 352.466903 
## iter  10 value 265.296842
## iter  20 value 259.530895
## iter  30 value 258.505815
## iter  40 value 258.481781
## iter  50 value 258.481314
## iter  50 value 258.481312
## iter  50 value 258.481312
## final  value 258.481312 
## converged

summary(model_1c5)

## Call:
## multinom(formula = food ~ lake + size + gender + lake * gender, 
##     data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan    lakeOkl    lakeTra  size<=2.3   genderMale
## Bird   -16.263542 15.4631991 16.0110863 16.5768526 -1.1908325  14.87784409
## Inver   -1.046837 -2.1160620  0.8808395  0.5136922  1.5215173  -0.94732194
## Other   -1.681967  0.5689054  1.3192386  2.4229579 -0.0522542   0.01106523
## Rept    -2.318943  0.6134287  1.3720043  1.7963644 -0.1881737 -14.50610070
##       lakeHan:genderMale lakeOkl:genderMale lakeTra:genderMale
## Bird         -15.8646026        -33.5355368         -16.335073
## Inver          0.5034982          0.3048162           1.093681
## Other          0.6840401         -2.3514491          -1.347638
## Rept          -7.7570090         14.5535117          14.612770
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    0.6627872 0.6452407 1.1230758 0.9794898 0.6863823  0.5932994
## Inver   0.5534699 0.7863163 0.8191284 0.9686761 0.4618276  0.6126551
## Other   0.7978770 0.9140771 1.2081530 1.1815357 0.4926099  0.9471840
## Rept    1.0821756 1.2371499 1.5813617 1.6465079 0.6981059  0.6905281
##       lakeHan:genderMale lakeOkl:genderMale lakeTra:genderMale
## Bird        1.124843e+00       2.393237e-07          1.1965529
## Inver       1.369324e+00       1.085099e+00          1.1800008
## Other       1.167883e+00       1.731244e+00          1.4487918
## Rept        9.243461e-10       9.573608e-01          0.9861938
## 
## Residual Deviance: 516.9626 
## AIC: 588.9626

# model Y~X1+X2+X3+X2X3
model_1c6 <- multinom(food ~ lake + size + gender +size*gender,data=datafood)

## # weights:  40 (28 variable)
## initial  value 352.466903 
## iter  10 value 269.657367
## iter  20 value 267.859275
## iter  30 value 267.849421
## iter  30 value 267.849420
## iter  30 value 267.849420
## final  value 267.849420 
## converged

summary(model_1c6)

## Call:
## multinom(formula = food ~ lake + size + gender + size * gender, 
##     data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan    lakeOkl  lakeTra  size<=2.3 genderMale
## Bird    -1.610300  0.6067870 -0.4587223 1.325001 -0.9499501 -0.8219859
## Inver   -1.101382 -1.7722719  0.9424647 1.184062  1.2416008 -0.5742209
## Other   -1.485725  0.8147021  0.1679189 1.684913 -0.1051924 -0.6960225
## Rept    -3.339082  0.9879282  2.2335485 2.778320  0.3511508  0.2037592
##       size<=2.3:genderMale
## Bird             0.4491087
## Inver            0.1497835
## Other            0.6635604
## Rept            -1.6862430
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    0.8007333 0.8014123 1.2462166 0.9110709 0.9163585  0.9299201
## Inver   0.6924399 0.6288826 0.5021787 0.5131789 0.7569443  0.7911095
## Other   0.6865768 0.5726299 0.8064032 0.6559994 0.7136479  0.7714181
## Rept    1.3673993 1.2035947 1.1455785 1.1486470 1.1892358  1.1845946
##       size<=2.3:genderMale
## Bird             1.3372394
## Inver            0.9344216
## Other            0.9506783
## Rept             1.6487195
## 
## Residual Deviance: 535.6988 
## AIC: 591.6988

# model Y~X1+X2+X3+X1X2+X1X3
model_1c7 <-multinom(food~lake+size+gender+lake*size+lake*gender, data=datafood)

## # weights:  65 (48 variable)
## initial  value 352.466903 
## iter  10 value 263.550767
## iter  20 value 252.470318
## iter  30 value 249.416728
## iter  40 value 249.182771
## iter  50 value 249.171406
## iter  60 value 249.171094
## final  value 249.171077 
## converged

summary(model_1c7)

## Warning in sqrt(diag(vc)): NaNs produced

## Call:
## multinom(formula = food ~ lake + size + gender + lake * size + 
##     lake * gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan    lakeOkl   lakeTra   size<=2.3   genderMale
## Bird   -26.545198  26.052061  28.139480 27.156201  0.35746095  24.32889436
## Inver   -2.325022 -11.825325   3.343036  2.464092  3.39642459  -1.34474341
## Other   -1.723879   1.046963 -24.458241  2.707398  0.06578482  -0.01984412
## Rept   -23.134727  21.806167  23.280548 23.102673 21.98323898 -20.17051634
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3 lakeHan:genderMale
## Bird         -2.0599485         -41.07881        -1.8707416        -25.3707049
## Inver         9.1294547          -3.19803        -2.5823130          0.9185007
## Other        -0.7054789          25.78133        -0.3803866          0.6513613
## Rept        -22.6845546         -23.42510       -22.7124870         -6.5400092
##       lakeOkl:genderMale lakeTra:genderMale
## Bird         -49.7462758         -25.988068
## Inver         -0.2158311           1.066904
## Other         -0.4355696          -1.489857
## Rept          19.2273535          19.910052
## 
## Std. Errors:
##       (Intercept)    lakeHan   lakeOkl  lakeTra size<=2.3 genderMale
## Bird    0.7760831   0.958557 1.6265326 1.492307 1.2943109  0.8719357
## Inver   1.0551636 487.518218 1.5826690 1.569481 1.1416983  0.7209299
## Other   0.8169634   1.077639 0.6805341 1.564129 0.9248934  0.9809570
## Rept    0.8301478   1.142391 1.6281559 1.528664 0.8301478  0.8056223
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3 lakeHan:genderMale
## Bird           1.653312               NaN          2.189651       1.305838e+00
## Inver        487.518527         1.5483507          1.422054       1.423678e+00
## Other          1.175912         0.6805341          1.434644       1.195075e+00
## Rept           1.309035         1.5503497          1.188362       2.081841e-10
##       lakeOkl:genderMale lakeTra:genderMale
## Bird                 NaN           1.525571
## Inver           1.336203           1.281557
## Other           1.786260           1.589410
## Rept            1.339247           1.204804
## 
## Residual Deviance: 498.3422 
## AIC: 594.3422

# model Y~X1+X2+X3+X1X2+X2X3
model_1c8 <-multinom(food~lake+size+gender+lake*size+size*gender,data=datafood)

## # weights:  55 (40 variable)
## initial  value 352.466903 
## iter  10 value 264.007344
## iter  20 value 258.635097
## iter  30 value 257.471456
## iter  40 value 257.268765
## iter  50 value 257.260181
## iter  60 value 257.260007
## iter  60 value 257.260005
## iter  60 value 257.260005
## final  value 257.260005 
## converged

summary(model_1c8)

## Call:
## multinom(formula = food ~ lake + size + gender + lake * size + 
##     size * gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan     lakeOkl   lakeTra   size<=2.3  genderMale
## Bird    -2.374902   1.905770   0.7777711  2.380109  0.78737738 -1.00313562
## Inver   -2.089062 -12.450905   3.7710436  4.210933  2.65814738 -2.34599930
## Other   -1.684653   1.390753 -22.9574759  1.301454  0.04589109 -0.08759659
## Rept   -26.119021  24.440030  25.9644059 26.459374 24.29284507 -0.63691750
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3
## Bird         -2.5432412         -32.94969        -2.0141052
## Inver        10.3127719          -3.39441        -3.6750412
## Other        -0.7904554          24.10091         0.3663241
## Rept        -24.6885868         -25.43077       -24.8435013
##       size<=2.3:genderMale
## Bird            0.25393402
## Inver           1.75745447
## Other           0.03952345
## Rept           -1.13638396
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    1.1116398 1.2505257 1.6186528 1.4283751  1.470820  1.1460683
## Inver   1.0663058 0.3359278 1.5774330 1.6200467  1.167156  1.3417232
## Other   0.7843509 0.8599831 0.4963547 0.9195573  1.090735  0.8394745
## Rept    1.1177932 1.0621017 0.9635959 0.9918237  1.145486  1.5602027
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3
## Bird          1.6767817      1.073912e-13          1.957890
## Inver         0.3359258      1.713893e+00          1.743582
## Other         1.1615493      4.963547e-01          1.283286
## Rept          1.2443446      1.359291e+00          1.229240
##       size<=2.3:genderMale
## Bird              1.533010
## Inver             1.425687
## Other             1.019876
## Rept              1.957142
## 
## Residual Deviance: 514.52 
## AIC: 594.52

# model Y~X1+X2+X3+X1X3+X2X3
model_1c9 <-multinom(food~lake+size+gender+lake*gender+size*gender,data=datafood)

## # weights:  55 (40 variable)
## initial  value 352.466903 
## iter  10 value 264.575413
## iter  20 value 258.989840
## iter  30 value 257.785301
## iter  40 value 257.722664
## iter  50 value 257.721130
## final  value 257.721117 
## converged

summary(model_1c9)

## Call:
## multinom(formula = food ~ lake + size + gender + lake * gender + 
##     size * gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan    lakeOkl    lakeTra  size<=2.3  genderMale
## Bird  -16.2454250 15.6026638 16.1814396 16.8103429 -1.4910019  14.7243894
## Inver  -0.9882986 -2.1039274  0.8933136  0.5292055  1.4356770  -0.9763839
## Other  -1.5457745  0.6835996  1.4561166  2.5972179 -0.3942057  -0.2679747
## Rept   -2.6795341  0.3888810  1.1089359  1.4747671  0.5445618 -13.0797407
##       lakeHan:genderMale lakeOkl:genderMale lakeTra:genderMale
## Bird         -16.0142292        -32.7287883         -16.454575
## Inver          0.5012999          0.2534458           1.043671
## Other          0.5702386         -2.3683123          -1.427343
## Rept          -6.8055922         13.7953320          13.964518
##       size<=2.3:genderMale
## Bird            0.60936071
## Inver           0.06734568
## Other           0.55921244
## Rept           -1.40622285
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    0.7021724 0.6592361 1.1454332 1.0305022 0.9622487  0.7992739
## Inver   0.6928949 0.8027431 0.8333448 0.9877422 0.7899435  0.9277684
## Other   0.8223479 0.9407293 1.2384019 1.2313676 0.7728513  1.0714394
## Rept    1.2585713 1.2622550 1.6052792 1.6755239 1.2195085  0.9643344
##       lakeHan:genderMale lakeOkl:genderMale lakeTra:genderMale
## Bird        1.129073e+00       5.670007e-07           1.206149
## Inver       1.379981e+00       1.082868e+00           1.178162
## Other       1.189446e+00       1.736694e+00           1.464333
## Rept        3.891958e-09       1.001575e+00           1.058997
##       size<=2.3:genderMale
## Bird             1.3510612
## Inver            0.9707251
## Other            0.9933454
## Rept             1.6718032
## 
## Residual Deviance: 515.4422 
## AIC: 595.4422

# model Y~X1+X2+X3+X1X2+X1X3+X2X3
model_1c10 <-multinom(food~lake+size+gender+lake*size+lake*gender+size*gender, data=datafood)

## # weights:  70 (52 variable)
## initial  value 352.466903 
## iter  10 value 261.806815
## iter  20 value 249.874155
## iter  30 value 245.251330
## iter  40 value 244.794341
## iter  50 value 244.772299
## iter  60 value 244.771293
## final  value 244.771286 
## converged

summary(model_1c10)

## Call:
## multinom(formula = food ~ lake + size + gender + lake * size + 
##     lake * gender + size * gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird   -29.680243  29.103199  68.77201 39.611336 13.866329  27.422556
## Inver   -2.006630 -14.493190  41.09846 40.176780  2.933473 -39.756994
## Other   -1.560861   1.181926 -38.81781  3.731800 -0.534904  -0.333421
## Rept   -34.044627  32.774394  21.58248  2.350668 32.774659 -14.981245
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3 lakeHan:genderMale
## Bird        -15.3329775         -79.63494        -24.490655        -28.0806124
## Inver        11.9281761         -40.92665        -40.410530          0.7598849
## Other        -0.5527564          40.50818         -0.942941          0.4179860
## Rept        -33.5480948         -21.41115         -1.774053        -14.3102392
##       lakeOkl:genderMale lakeTra:genderMale size<=2.3:genderMale
## Bird         -88.1781781         -38.334497          -13.4362543
## Inver          0.0461318           1.299149           38.6120194
## Other         -0.8036763          -2.307518            0.8491819
## Rept          26.6702270          46.387523          -31.8115065
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    0.7070832 0.8084083 1.4658431 0.5917075 0.5509420  0.7777457
## Inver   1.0638255 0.4393419 1.0245550 1.2212504 1.2249471  0.9993125
## Other   0.8276239 1.0697093 0.7768042 2.0107535 1.2926849  1.0971195
## Rept    0.7485178 0.9206041 0.7360161 0.9455311 0.7553727  0.4517334
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3 lakeHan:genderMale
## Bird          0.8905558      5.432772e-11         0.7160007       1.237839e+00
## Inver         0.4393416      9.390471e-01         0.9931620       1.433917e+00
## Other         1.2294201      7.768042e-01         1.6393334       1.283608e+00
## Rept          1.0822488      9.218556e-01         0.8925133       4.926739e-13
##       lakeOkl:genderMale lakeTra:genderMale size<=2.3:genderMale
## Bird        3.296180e-08          0.8287452            1.0419266
## Inver       1.413217e+00          1.3337518            0.9072072
## Other       1.997547e+00          1.9339587            1.1962097
## Rept        4.255527e-01          0.5428352            0.9846263
## 
## Residual Deviance: 489.5426 
## AIC: 593.5426

# model 
Y~X1+X2+X3+X1X2+X1X3+X2X3+X1X2X3

## Y ~ X1 + X2 + X3 + X1X2 + X1X3 + X2X3 + X1X2X3

model_1c11 <-multinom(food~lake+size+gender+lake*size+lake*gender+size*gender+lake*size*gender,
data=datafood)

## # weights:  85 (64 variable)
## initial  value 352.466903 
## iter  10 value 260.145098
## iter  20 value 247.220833
## iter  30 value 244.162621
## iter  40 value 243.815027
## iter  50 value 243.801406
## iter  60 value 243.800899
## iter  60 value 243.800898
## iter  60 value 243.800898
## final  value 243.800898 
## converged

summary(model_1c11)

## Call:
## multinom(formula = food ~ lake + size + gender + lake * size + 
##     lake * gender + size * gender + lake * size * gender, data = datafood)
## 
## Coefficients:
##       (Intercept)    lakeHan    lakeOkl   lakeTra  size<=2.3 genderMale
## Bird   -23.659137  23.253776  45.302701 18.666966 -5.1636921  21.461986
## Inver   -2.079465 -12.135591  23.723299 41.083525  3.1778456 -32.399336
## Other   -2.079548   2.079581 -19.265751 -6.158001  0.9806851   0.575526
## Rept   -29.481972  28.383330   8.903792  8.739972 28.3828917  -9.045851
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3 lakeHan:genderMale
## Bird           3.489575        -44.628018          9.462975         -22.443051
## Inver          9.363311        -23.723138        -41.488498          -0.378315
## Other         -2.654656         19.959036          7.950245          -1.268768
## Rept         -29.363666         -8.903379         -8.333677         -13.270796
##       lakeOkl:genderMale lakeTra:genderMale size<=2.3:genderMale
## Bird           -68.65129         -17.450651             5.489023
## Inver           10.13646          -6.892404            31.038624
## Other          -14.17904           7.192033            -1.348468
## Rept            28.85082          29.500163           -21.726694
##       lakeHan:size<=2.3:genderMale lakeOkl:size<=2.3:genderMale
## Bird                    -29.234803                    0.4598397
## Inver                     1.466961                   -9.8742203
## Other                     3.379153                   14.6644729
## Rept                      4.422985                  -19.9036846
##       lakeTra:size<=2.3:genderMale
## Bird                   -44.1067826
## Inver                    8.4069123
## Other                   -8.2112240
## Rept                     0.8663284
## 
## Std. Errors:
##       (Intercept)   lakeHan   lakeOkl   lakeTra size<=2.3 genderMale
## Bird    0.7076713 0.8055881 1.2170267 0.5899784 0.5605073  0.7643122
## Inver   1.0606630 0.4583132 0.6686450 0.6394553 1.2527514  0.6124146
## Other   1.0607033 1.3385692 0.7359783 0.9350246 1.5679312  1.3176434
## Rept    0.7713307 0.9140028 0.7471843 0.8686971 0.7713308  0.4213413
##       lakeHan:size<=2.3 lakeOkl:size<=2.3 lakeTra:size<=2.3 lakeHan:genderMale
## Bird          0.9039139      9.842819e-12         0.7030400       1.229372e+00
## Inver         0.4583076      8.466216e-01         0.7271145       7.348490e-01
## Other         1.8764086      7.359783e-01         0.9350246       1.775630e+00
## Rept          1.0804467      9.296716e-01         0.9101667       9.906137e-10
##       lakeOkl:genderMale lakeTra:genderMale size<=2.3:genderMale
## Bird        4.851911e-10          0.8268938            1.0437480
## Inver       6.900638e-01          0.7756274            0.7546887
## Other       1.029998e+00          0.9350246            1.9095632
## Rept        4.222621e-01          0.4827984            0.5984610
##       lakeHan:size<=2.3:genderMale lakeOkl:size<=2.3:genderMale
## Bird                  9.598235e-11                 9.920193e-15
## Inver                 7.348490e-01                 9.390853e-01
## Other                 2.408691e+00                 1.029998e+00
## Rept                  9.081705e-18                 4.538504e-10
##       lakeTra:size<=2.3:genderMale
## Bird                  1.471557e-14
## Inver                 8.139676e-01
## Other                 1.352559e+00
## Rept                  5.984610e-01
## 
## Residual Deviance: 487.6018 
## AIC: 615.6018

# Melihat deviance, derajat bebas dan AIC
# Model Y~X1
model_1c1$deviance

## [1] 561.1677

model_1c1$edf

## [1] 16

model_1c1$AIC

## [1] 593.1677

# Model Y~X2
model_1c2$deviance

## [1] 589.2134

model_1c2$edf

## [1] 8

model_1c2$AIC

## [1] 605.2134

# Model Y~X3
model_1c3$deviance

## [1] 602.2589

model_1c3$edf

## [1] 8

model_1c3$AIC

## [1] 618.2589

# Model Y~X1+X2
model_1$deviance

## [1] 540.0803

model_1$edf

## [1] 20

model_1$AIC

## [1] 580.0803

# Model Y~X1+X3
model_3$deviance

## [1] 555.4658

model_3$edf

## [1] 20

model_3$AIC

## [1] 595.4658

# Model Y~X2+X3
model_4$deviance

## [1] 588.1835

model_4$edf

## [1] 12

model_4$AIC

## [1] 612.1835

# Model Y~X1+X2+X3
model_2$deviance

## [1] 537.8655

model_2$edf

## [1] 24

model_2$AIC

## [1] 585.8655

# Model Y~X1+X2+X3+X1X2
model_1c4$deviance

## [1] 517.6379

model_1c4$edf

## [1] 36

model_1c4$AIC

## [1] 589.6379

# Model Y~X1+X2+X3+X1X3
model_1c5$deviance

## [1] 516.9626

model_1c5$edf

## [1] 36

model_1c5$AIC

## [1] 588.9626

# Model Y~X1+X2+X3+X2X3
model_1c6$deviance

## [1] 535.6988

model_1c6$edf

## [1] 28

model_1c6$AIC

## [1] 591.6988

# Model Y~X1+X2+X3+X1X2+X1X3
model_1c7$deviance

## [1] 498.3422

model_1c7$edf

## [1] 48

model_1c7$AIC

## [1] 594.3422

# Model Y~X1+X2+X3+X1X2+X2X3
model_1c8$deviance

## [1] 514.52

model_1c8$edf

## [1] 40

model_1c8$AIC

## [1] 594.52

# Model Y~X1+X2+X3+X1X3+X2X3
model_1c9$deviance

## [1] 515.4422

model_1c9$edf

## [1] 40

model_1c9$AIC

## [1] 595.4422

# Model Y~X1+X2+X3+X1X2+X1X3+X2X3
model_1c10$deviance

## [1] 489.5426

model_1c10$edf

## [1] 52

model_1c10$AIC

## [1] 593.5426

# Model Y~X1+X2+X3+X1X2+X1X3+X2X3+X1X2X3
model_1c11$deviance

## [1] 487.6018

model_1c11$edf

## [1] 64

model_1c11$AIC

## [1] 615.6018

Ringkasan berdasarkan hasil output:

Ringkasan hasil uji deviance:

Berdasarkan pengujian poin (d) dan poin (b), dapat disimpulkan bahwa model terbaik adalah model dengan peubah bebas size dan lake (Model 4). Hal ini juga dapat dipastikan dengan nilai AIC yang lebih kecil dibandingkan dengan model lainnya.

Soal 2

A. Persamaan Prediksi untuk Log(phi1/phi2)

B. Dengan menggunakan kategori ya dan tidak, interpretasikan efek bersyarat gender menggunakanan Selang kepercayaan 95% untuk odds ratio

C. Tunjukan bahwa jika seseorang itu wanita dan putih maka Phi_cap_1= P_cap(Y=yes)=0,76

D

E

F

Mahasiswa Pascasarjana Statistika dan Sains Data, IPB University, reniamelia@apps.ipb.ac.id ↩︎