-以每年參觀 Lake Keepit 的人數為例

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-以每年參觀 Lake Keepit 的人數為例

指導老師：蔡碧紋學生：林柏佐系級：數學系 98 級

班別：四年乙班學號：494402345

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>

lake=read.table("D:lake.txt",header=T)

> lake

#Dist=Distance, Inc=Family Income, Size=Family Members,

Y=Numbers of Vistors Dist Inc Size Y 1 27 4.45 5 1 2 72 7.69 4 1 3 44 10.04 4 5 4 23 8.97 4 4 5 89 9.15 5 2 6 58 5.80 5 5 7 84 4.94 4 2 8 62 5.05 1 1 9 10 2.89 4 8 10 77 4.21 3 0 11 91 2.93 3 0 12 72 5.46 5 0 13 54 7.98 2 3 14 34 7.39 1 6 15 94 4.71 4 1 16 98 6.55 4 0 17 90 3.61 3 0 18 63 10.42 4 0 19 19 7.36 2 5 20 34 2.67 3 1 21 45 3.76 4 5 22 91 8.71 5 1 23 40 6.32 3 3 24 78 5.64 2 4 25 76 7.51 2 0 26 64 7.24 5 5 27 62 9.13 1 1 28 20 7.98 4 3 29 119 3.61 5 0 30 50 3.21 3 2 31 81 6.55 1 1

32 38 2.66 2 3 33 29 6.68 4 3 34 100 5.60 1 1 35 78 9.69 5 0 36 46 7.82 5 4 37 56 7.34 2 0 38 34 5.39 1 2 39 63 8.36 5 4 40 34 5.95 1 1 41 120 6.86 1 1 42 85 9.50 2 1 43 25 8.00 2 2 44 37 8.29 5 8 45 21 8.57 4 7 46 88 6.29 3 0 47 66 5.08 1 2 48 48 5.82 2 1 49 90 7.39 2 1 50 57 6.61 4 2 51 109 9.88 2 0 52 57 7.96 3 2 53 19 3.86 2 1 54 79 3.15 3 0 55 30 9.64 3 5 56 48 3.56 2 3 57 41 6.18 2 0 58 49 5.79 2 3 59 59 6.89 2 0 60 72 2.74 2 2 61 76 8.85 5 3 62 110 9.83 4 0 63 47 6.38 1 0 64 88 7.47 3 0 65 10 5.39 4 3 66 21 10.54 5 7 67 64 10.39 3 2 68 46 7.89 4 4 69 44 5.40 2 4

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70 79 3.74 4 3 71 116 6.02 2 0 72 84 2.93 3 1 73 95 10.29 5 2 74 17 6.68 4 7 75 98 2.76 4 0 76 91 4.21 5 6 77 86 3.81 5 2 78 118 3.95 3 0 79 85 10.50 2 1 80 87 6.93 1 1 81 53 8.89 1 2 82 70 2.69 1 0 83 44 9.94 2 6 84 54 6.77 5 4 85 42 6.73 3 4 86 27 3.07 5 6 87 62 9.90 2 0 88 57 3.96 2 1 89 117 8.33 4 0 90 101 5.97 1 1 91 94 7.42 3 2 92 46 4.25 1 5 93 84 8.84 2 2 94 20 4.02 1 5 95 102 5.33 3 0 96 50 9.24 5 5 97 52 10.53 2 4 98 52 4.51 4 6 99 100 5.18 3 1 100 84 8.77 1 3 101 35 7.45 5 5 102 78 9.20 4 0 103 90 9.50 5 0 104 97 7.19 5 0 105 49 6.69 3 0 106 43 3.18 5 4 107 96 5.48 1 0

108 37 6.09 5 4 109 65 9.55 2 1 110 50 7.46 4 7 111 34 3.60 3 4 112 108 9.68 4 1 113 20 3.78 5 5 114 77 8.77 3 0 115 86 7.53 2 1 116 106 5.45 1 0 117 20 4.08 5 2 118 29 7.61 3 3 119 102 8.68 4 2 120 45 8.73 2 2 121 87 9.52 1 3 122 18 3.72 3 7 123 96 4.63 4 1 124 38 8.10 4 4 125 54 9.13 3 4 126 42 2.75 2 2 127 105 10.59 3 2 128 12 4.44 3 9 129 74 9.53 1 1 130 108 8.96 5 0 131 12 7.33 3 5 132 90 5.88 3 0 133 28 9.08 2 2 134 102 3.42 5 0 135 45 5.87 1 1 136 45 6.15 4 1 137 112 7.77 1 0 138 83 8.10 4 1 139 35 2.96 3 3 140 80 8.34 5 1 141 104 5.13 5 0 142 78 7.24 4 3 143 55 3.19 2 0 144 25 4.20 5 2 145 78 4.02 4 0

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146 87 8.78 5 3 147 111 8.85 3 2 148 36 10.01 1 4 149 116 8.12 5 0 150 36 3.84 3 6 151 104 6.88 5 2 152 60 3.79 1 4 153 114 8.42 1 2 154 56 6.95 2 1 155 66 4.33 5 2 156 65 5.34 4 4 157 105 5.95 2 0 158 66 9.90 2 0 159 94 9.96 4 1 160 113 6.03 2 0 161 93 9.04 3 1 162 51 8.13 3 3 163 87 5.35 2 1 164 22 10.09 4 5 165 46 4.88 4 0 166 99 2.63 4 2 167 14 5.99 4 3 168 14 10.25 5 10 169 89 9.14 2 1 170 103 6.55 4 1 171 20 4.24 3 5 172 32 10.21 2 3 173 109 5.90 5 1 174 55 8.39 5 3 175 94 3.61 2 2 176 64 8.40 3 1 177 110 4.41 1 1 178 99 3.69 3 0 179 36 9.05 1 3 180 46 10.31 2 4 181 115 2.87 4 0 182 25 9.90 2 5 183 63 2.87 4 2

184 94 6.45 1 1 185 24 9.64 3 2 186 73 5.92 4 1 187 25 6.06 1 6 188 91 4.31 5 2 189 81 3.54 4 5 190 118 7.32 4 0 191 86 4.20 2 0 192 86 6.98 1 0 193 97 6.93 5 4 194 41 6.72 3 3 195 77 4.01 1 0 196 111 6.81 2 0 197 70 7.94 1 0 198 66 3.36 2 1 199 30 3.69 5 5 200 115 3.38 4 0 201 12 7.46 3 3 202 100 7.61 1 0 203 71 9.22 1 1 204 37 10.43 1 1 205 35 3.78 4 3 206 110 3.68 5 2 207 76 8.01 5 1 208 119 6.30 5 2 209 70 6.39 1 0 210 104 5.74 2 0 211 38 9.91 2 5 212 70 6.04 4 2 213 88 6.51 4 1 214 30 5.82 5 4 215 76 7.18 1 2 216 73 4.91 5 0 217 54 8.27 5 5 218 28 5.09 3 4 219 64 6.74 4 2 220 28 4.99 1 3 221 37 6.41 1 1

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222 19 9.14 3 4 223 59 7.78 3 1 224 102 7.71 4 0 225 34 9.62 3 2 226 14 4.20 5 7 227 88 10.37 1 1 228 113 7.22 5 0 229 72 5.81 4 1 230 42 4.72 3 2 231 18 9.30 2 4 232 58 5.40 5 3 233 91 5.98 4 0 234 67 8.86 4 1 235 94 4.35 4 3 236 39 3.30 5 2 237 79 8.16 4 1 238 103 9.16 2 3 239 91 8.24 5 1 240 101 9.41 4 2 241 59 7.58 2 0 242 25 4.52 3 3 243 11 7.37 3 7 244 21 8.85 3 6 245 40 5.86 5 1 246 23 3.94 4 13 247 58 10.07 1 1 248 24 8.99 1 3 249 32 10.10 4 3 250 118 4.14 2 2

> attach(lake)

> pairs(lake)

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從圖形來看，距離與人數有很高的 correlation

> cor(lake)

Dist Inc Size Y Dist 1.00000000 -0.00382533 0.02006100 -0.62618753 Inc -0.00382533 1.00000000 -0.08055719 0.01823499 Size 0.02006100 -0.08055719 1.00000000 0.18961580 Y -0.62618753 0.01823499 0.18961580 1.00000000

> fm<-lm(Y~Dist+Inc+Size)

> summary(fm)

Call:

lm(formula = Y ~ Dist + Inc + Size)

Residuals:

Min 1Q Median 3Q Max -3.4949 -1.0841 -0.1381 1.0107 8.6605

Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 3.973939 0.468093 8.490 1.98e-15 ***

Dist -0.044974 0.003424 -13.136 < 2e-16 ***

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Inc 0.031041 0.046206 0.672 0.502#無法 reject Size 0.319426 0.075047 4.256 2.96e-05 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.652 on 246 degrees of freedom

Multiple R-squared: 0.434, Adjusted R-squared: 0.4271#太低了 F-statistic: 62.89 on 3 and 246 DF, p-value: < 2.2e-16

> anova(fm)

Analysis of Variance Table

Response: Y

Df Sum Sq Mean Sq F value Pr(>F) Dist 1 465.29 465.29 170.4352 < 2.2e-16 ***

Inc 1 0.30 0.30 0.1091 0.7415#還是收入有問題 Size 1 49.46 49.46 18.1165 2.957e-05 ***

Residuals 246 671.58 2.73 ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> par(mfrow=c(2,2))

> plot(fm,ask=F)

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由圖形來看，residuals 有一個 pattern 在，因此考慮是否要做 transformation

> boxcox(fm)

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> boxcox(fm,lambda=seq(-0.5,0.5,0.1))

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> fm1<-lm(log(Y+1)~Dist+Inc+Size)#在做轉換時，要注意各係數都必須是正數，

因為 Y 有 0，所以我讓其加 1，來做 regression

> summary(fm1)

Call:

lm(formula = log(Y + 1) ~ Dist + Inc + Size)

Residuals:

Min 1Q Median 3Q Max -1.27011 -0.34697 0.04571 0.36514 1.23818

Coefficients:

Estimate Std. Error t value Pr(>|t|) (Intercept) 1.47987 0.14623 10.120 < 2e-16 ***

Dist -0.01415 0.00107 -13.234 < 2e-16 ***

Inc 0.01892 0.01443 1.310 0.191285#還是 Income 有問題 Size 0.08727 0.02345 3.722 0.000245 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

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Residual standard error: 0.5162 on 246 degrees of freedom Multiple R-squared: 0.4335, Adjusted R-squared: 0.4266 F-statistic: 62.75 on 3 and 246 DF, p-value: < 2.2e-16

> anova(fm1)

Response: log(Y + 1)

Inc 1 0.274 0.274 1.0283 0.3115555 Size 1 3.692 3.692 13.8554 0.0002447 ***

Residuals 246 65.542 0.266 ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> par(mfrow=c(2,2))

>

plot(fm1,ask=F)

來做 model selection

> b=leaps(cbind(Dist,Inc,Size),log(Y+1),method="r2")

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> b.r2=b$r2

> b=leaps(cbind(Dist,Inc,Size),log(Y+1),method="adjr2")

> b.adjr2=b$adjr2

> b=leaps(cbind(Dist,Inc,Size),log(Y+1),method="Cp")

> b.Cp=b$Cp

> plot(b$size,b.Cp)

1 2 3 b.r2 b.adjr2 b.Cp 1 1 0 0 0.399220586 0.396798088 14.883715 1 0 0 1 0.026055915 0.022128721 176.927526 1 0 1 0 0.002609064 -0.001412674 187.109136 2 1 0 1 0.429541430 0.424922332 3.717128 2 1 1 0 0.401588611 0.396743175 15.855419 2 0 1 1 0.030189294 0.022336576 177.132639 3 1 1 1 0.433495741 0.426587152 4.000000

> bs=regsubsets(log(Y+1)~Dist+Inc+Size,data=lake)

> rs=summary(bs)

> names(rs)

[1] "which" "rsq" "rss" "adjr2" "cp" "bic" "outmat"

[8] "obj"

> print(cbind(rs$which,R2=rs$rsq,adjR2=rs$adjr2,Cp=rs$cp,BIC=rs$bic))

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(Intercept) Dist Inc Size R2 adjR2 Cp BIC 1 1 1 0 0 0.3992206 0.3967981 14.883715 -116.3389 2 1 1 0 1 0.4295414 0.4249223 3.717128 -123.7643 3 1 1 1 1 0.4334957 0.4265872 4.000000 -119.9818

> fm2<-lm(log(Y+1)~Dist+Size)

> summary(fm2)

Call:

lm(formula = log(Y + 1) ~ Dist + Size)

Residuals:

Min 1Q Median 3Q Max -1.30145 -0.33937 0.04727 0.36763 1.19679

Coefficients:

Dist -0.014158 0.001071 -13.218 < 2e-16 ***

Size 0.084795 0.023403 3.623 0.000353 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> anova(fm2)

Response: log(Y + 1)

Size 1 3.508 3.508 13.129 0.000353 ***

Residuals 247 66.000 0.267 ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> par(mfrow=c(2,2))

> plot(fm2,ask=F)

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> fm3<-lm(sqrt(Y)~Dist+Inc+Size) #variance stable

> summary(fm3)

Call:

lm(formula = sqrt(Y) ~ Dist + Inc + Size)

Residuals:

Min 1Q Median 3Q Max -1.6186 -0.4273 0.0727 0.4570 1.6070

Coefficients:

Dist -0.017528 0.001348 -13.000 < 2e-16 ***

Inc 0.024612 0.018196 1.353 0.17742 Size 0.104933 0.029553 3.551 0.00046 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6507 on 246 degrees of freedom

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Multiple R-squared: 0.424, Adjusted R-squared: 0.417 F-statistic: 60.36 on 3 and 246 DF, p-value: < 2.2e-16

> anova(fm3)

Response: sqrt(Y)

Inc 1 0.485 0.485 1.1455 0.2855408 Size 1 5.337 5.337 12.6072 0.0004603 ***

Residuals 246 104.145 0.423 ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> par(mfrow=c(2,2))

> plot(fm3,ask=F)

> res=residuals(fm3)

> fm4=lm(abs(res)~Dist+Inc+Size)

> wi=1/((fitted(fm4))^2)

> print(cbind(Dist,Inc,Size,Y,res,"abs(rs)"=abs(res),weights=wi),digits=2) Dist Inc Size Y res abs(rs) weights

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1 27 4.4 5 1 -1.0459 1.0459 4.2 2 72 7.7 4 1 -0.2320 0.2320 3.6 3 44 10.0 4 5 0.4554 0.4554 5.4 4 23 9.0 4 4 -0.1224 0.1224 6.0 5 89 9.2 5 2 0.3393 0.3393 3.5 6 58 5.8 5 5 0.7002 0.7002 3.6 7 84 4.9 4 2 0.4602 0.4602 2.9 8 62 5.0 1 1 -0.0275 0.0275 3.3 9 10 2.9 4 8 0.6278 0.6278 4.4 10 77 4.2 3 0 -0.9538 0.9538 2.9 11 91 2.9 3 0 -0.6769 0.6769 2.5 12 72 5.5 5 0 -1.2821 1.2821 3.2 13 54 8.0 2 3 0.3873 0.3873 4.2 14 34 7.4 1 6 0.8736 0.8736 4.8 15 94 4.7 4 1 0.2269 0.2269 2.6 16 98 6.5 4 0 -0.7482 0.7482 2.8 17 90 3.6 3 0 -0.7112 0.7112 2.6 18 63 10.4 4 0 -1.4570 1.4570 4.7 19 19 7.4 2 5 0.2931 0.2931 5.5 20 34 2.7 3 1 -0.6696 0.6696 3.6 21 45 3.8 4 5 0.6275 0.6275 3.5 22 91 8.7 5 1 -0.0290 0.0290 3.4 23 40 6.3 3 3 0.0778 0.0778 4.3 24 78 5.6 2 4 1.1335 1.1335 3.1 25 76 7.5 2 0 -0.9476 0.9476 3.5 26 64 7.2 5 5 0.7700 0.7700 3.8 27 62 9.1 1 1 -0.1279 0.1279 4.3 28 20 8.0 4 3 -0.4185 0.4185 5.7 29 119 3.6 5 0 -0.4127 0.4127 2.2 30 50 3.2 3 2 0.0118 0.0118 3.3 31 81 6.5 1 1 0.2686 0.2686 3.2 32 38 2.7 2 3 0.2378 0.2378 3.5 33 29 6.7 4 3 -0.2288 0.2288 4.8 34 100 5.6 1 1 0.6250 0.6250 2.7 35 78 9.7 5 0 -1.2810 1.2810 3.9 36 46 7.8 5 4 0.2041 0.2041 4.5 37 56 7.3 2 0 -1.2940 1.2940 4.0 38 34 5.4 1 2 -0.1124 0.1124 4.2 39 63 8.4 5 4 0.4888 0.4888 4.1 40 34 6.0 1 1 -0.5404 0.5404 4.4 41 120 6.9 1 1 0.9445 0.9445 2.5 42 85 9.5 2 1 0.1612 0.1612 3.7 43 25 8.0 2 2 -0.4394 0.4394 5.5 44 37 8.3 5 8 0.8632 0.8632 5.0 45 21 8.6 4 7 0.4982 0.4982 5.9 46 88 6.3 3 0 -0.8122 0.8122 3.0 47 66 5.1 1 2 0.4561 0.4561 3.2 48 48 5.8 2 1 -0.3968 0.3968 3.9 49 90 7.4 2 1 0.3007 0.3007 3.1 50 57 6.6 4 2 -0.0541 0.0541 3.8

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51 109 9.9 2 0 -0.4275 0.4275 3.2 52 57 8.0 3 2 0.0176 0.0176 4.1 53 19 3.9 2 1 -0.8568 0.8568 4.3 54 79 3.1 3 0 -0.8927 0.8927 2.7 55 30 9.6 3 5 0.3248 0.3248 5.9 56 48 3.6 2 3 0.3909 0.3909 3.4 57 41 6.2 2 0 -1.5283 1.5283 4.2 58 49 5.8 2 3 0.3535 0.3535 3.8 59 59 6.9 2 0 -1.2303 1.2303 3.8 60 72 2.7 2 2 0.5139 0.5139 2.7 61 76 8.8 5 3 0.4367 0.4367 3.8 62 110 9.8 4 0 -0.6186 0.6186 3.1 63 47 6.4 1 0 -1.3232 1.3232 4.0 64 88 7.5 3 0 -0.8412 0.8412 3.2 65 10 5.4 4 3 -0.5301 0.5301 5.2 66 21 10.5 5 7 0.3448 0.3448 7.0 67 64 10.4 3 2 0.0805 0.0805 4.6 68 46 7.9 4 4 0.3073 0.3073 4.5 69 44 5.4 2 4 0.5434 0.5434 3.9 70 79 3.7 4 3 0.7199 0.7199 2.8 71 116 6.0 2 0 -0.2098 0.2098 2.5 72 84 2.9 3 1 0.2004 0.2004 2.6 73 95 10.3 5 2 0.4164 0.4164 3.6 74 17 6.7 4 7 0.4746 0.4746 5.3 75 98 2.8 4 0 -0.6550 0.6550 2.3 76 91 4.2 5 6 1.5312 1.5312 2.6 77 86 3.8 5 2 0.4181 0.4181 2.7 78 118 4.0 3 0 -0.2288 0.2288 2.2 79 85 10.5 2 1 0.1365 0.1365 3.9 80 87 6.9 1 1 0.3644 0.3644 3.1 81 53 8.9 1 2 0.1344 0.1344 4.5 82 70 2.7 1 0 -0.8292 0.8292 2.8 83 44 9.9 2 6 0.8812 0.8812 5.3 84 54 6.8 5 4 0.3702 0.3702 3.9 85 42 6.7 3 4 0.3707 0.3707 4.3 86 27 3.1 5 6 0.4375 0.4375 3.9 87 62 9.9 2 0 -1.2518 1.2518 4.5 88 57 4.0 2 1 -0.1933 0.1933 3.2 89 117 8.3 4 0 -0.4590 0.4590 2.8 90 101 6.0 1 1 0.6334 0.6334 2.7 91 94 7.4 3 2 0.6794 0.6794 3.1 92 46 4.2 1 5 0.9478 0.9478 3.6 93 84 8.8 2 2 0.5741 0.5741 3.6 94 20 4.0 1 5 0.4977 0.4977 4.3 95 102 5.3 3 0 -0.5432 0.5432 2.6 96 50 9.2 5 5 0.4754 0.4754 4.8 97 52 10.5 2 4 0.5574 0.5574 5.1 98 52 4.5 4 6 0.9452 0.9452 3.5 99 100 5.2 3 1 0.4255 0.4255 2.6 100 84 8.8 1 3 0.9986 0.9986 3.5

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101 35 7.4 5 5 0.2565 0.2565 4.8 102 78 9.2 4 0 -1.1640 1.1640 3.8 103 90 9.5 5 0 -1.0660 1.0660 3.6 104 97 7.2 5 0 -0.8865 0.8865 3.0 105 49 6.7 3 0 -1.5056 1.5056 4.1 106 43 3.2 5 4 0.2658 0.2658 3.4 107 96 5.5 1 0 -0.4422 0.4422 2.7 108 37 6.1 5 4 0.0890 0.0890 4.3 109 65 9.6 2 1 -0.1906 0.1906 4.3 110 50 7.5 4 7 1.0338 1.0338 4.3 111 34 3.6 3 4 0.3075 0.3075 3.8 112 108 9.7 4 1 0.3500 0.3500 3.2 113 20 3.8 5 5 0.0839 0.0839 4.3 114 77 8.8 3 0 -1.0660 1.0660 3.7 115 86 7.5 2 1 0.2272 0.2272 3.2 116 106 5.4 1 0 -0.2662 0.2662 2.5 117 20 4.1 5 2 -0.7453 0.7453 4.4 118 29 7.6 3 3 -0.1467 0.1467 5.1 119 102 8.7 4 2 0.6837 0.6837 3.1 120 45 8.7 2 2 -0.1068 0.1068 4.8 121 87 9.5 1 3 1.0327 1.0327 3.6 122 18 3.7 3 7 0.6699 0.6699 4.3 123 96 4.6 4 1 0.2640 0.2640 2.6 124 38 8.1 4 4 0.1620 0.1620 4.9 125 54 9.1 3 4 0.5220 0.5220 4.6 126 42 2.8 2 2 -0.0122 0.0122 3.4 127 105 10.6 3 2 0.7942 0.7942 3.4 128 12 4.4 3 9 0.9013 0.9013 4.8 129 74 9.5 1 1 0.0726 0.0726 4.0 130 108 9.0 5 0 -0.7372 0.7372 3.0 131 12 7.3 3 5 0.0662 0.0662 5.9 132 90 5.9 3 0 -0.7670 0.7670 2.9 133 28 9.1 2 2 -0.4134 0.4134 5.8 134 102 3.4 5 0 -0.7060 0.7060 2.4 135 45 5.9 1 1 -0.3457 0.3457 4.0 136 45 6.2 4 1 -0.6674 0.6674 4.1 137 112 7.8 1 0 -0.2181 0.2181 2.8 138 83 8.1 4 1 -0.0493 0.0493 3.4 139 35 3.0 3 3 0.0729 0.0729 3.6 140 80 8.3 5 1 -0.2127 0.2127 3.6 141 104 5.1 5 0 -0.7131 0.7131 2.5 142 78 7.2 4 3 0.6163 0.6163 3.4 143 55 3.2 2 0 -1.2094 1.2094 3.2 144 25 4.2 5 2 -0.6606 0.6606 4.2 145 78 4.0 4 0 -1.0365 1.0365 2.8 146 87 8.8 5 3 0.6312 0.6312 3.5 147 111 8.8 3 2 0.9422 0.9422 2.9 148 36 10.0 1 4 0.3947 0.3947 5.7 149 116 8.1 5 0 -0.5763 0.5763 2.7 150 36 3.8 3 6 0.7862 0.7862 3.8

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151 104 6.9 5 2 0.6581 0.6581 2.8 152 60 3.8 1 4 0.9684 0.9684 3.1 153 114 8.4 1 2 1.2152 1.2152 2.8 154 56 7.0 2 1 -0.2844 0.2844 3.9 155 66 4.3 5 2 0.0548 0.0548 3.1 156 65 5.3 4 4 0.7031 0.7031 3.3 157 105 6.0 2 0 -0.4009 0.4009 2.6 158 66 9.9 2 0 -1.1817 1.1817 4.4 159 94 10.0 4 1 0.0977 0.0977 3.5 160 113 6.0 2 0 -0.2627 0.2627 2.5 161 93 9.0 3 1 0.2078 0.2078 3.4 162 51 8.1 3 3 0.2261 0.2261 4.4 163 87 5.3 2 1 0.2984 0.2984 2.9 164 22 10.1 4 5 0.0686 0.0686 6.6 165 46 4.9 4 0 -1.6186 1.6186 3.7 166 99 2.6 4 2 0.7800 0.7800 2.3 167 14 6.0 4 3 -0.4747 0.4747 5.2 168 14 10.2 5 10 0.7457 0.7457 7.3 169 89 9.1 2 1 0.2401 0.2401 3.5 170 103 6.5 4 1 0.3394 0.3394 2.7 171 20 4.2 3 5 0.2825 0.2825 4.4 172 32 10.2 2 3 -0.0532 0.0532 6.0 173 109 5.9 5 1 0.3556 0.3556 2.6 174 55 8.4 5 3 0.0799 0.0799 4.3 175 94 3.6 2 2 0.8781 0.8781 2.5 176 64 8.4 3 1 -0.2848 0.2848 4.0 177 110 4.4 1 1 0.8296 0.8296 2.4 178 99 3.7 3 0 -0.5554 0.5554 2.4 179 36 9.1 1 3 0.1504 0.1504 5.3 180 46 10.3 2 4 0.4576 0.4576 5.3 181 115 2.9 4 0 -0.3597 0.3597 2.1 182 25 9.9 2 5 0.3357 0.3357 6.3 183 63 2.9 4 2 0.1431 0.1431 2.9 184 94 6.4 1 1 0.4989 0.4989 2.9 185 24 9.6 3 2 -0.6022 0.6022 6.3 186 73 5.9 4 1 -0.1709 0.1709 3.3 187 25 6.1 1 6 0.7486 0.7486 4.7 188 91 4.3 5 2 0.4935 0.4935 2.6 189 81 3.5 4 5 1.2639 1.2639 2.7 190 118 7.3 4 0 -0.4166 0.4166 2.6 191 86 4.2 2 0 -0.6909 0.6909 2.7 192 86 7.0 1 0 -0.6544 0.6544 3.1 193 97 6.9 5 4 1.1199 1.1199 2.9 194 41 6.7 3 3 0.0855 0.0855 4.4 195 77 4.0 1 0 -0.7390 0.7390 2.8 196 111 6.8 2 0 -0.3169 0.3169 2.6 197 70 7.9 1 0 -0.9584 0.9584 3.7 198 66 3.4 2 1 -0.0207 0.0207 3.0 199 30 3.7 5 5 0.2614 0.2614 3.9 200 115 3.4 4 0 -0.3723 0.3723 2.2

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201 12 7.5 3 3 -0.4410 0.4410 5.9 202 100 7.6 1 0 -0.4245 0.4245 3.0 203 71 9.2 1 1 0.0276 0.0276 4.0 204 37 10.4 1 1 -0.5981 0.5981 5.9 205 35 3.8 4 3 -0.0523 0.0523 3.8 206 110 3.7 5 2 0.8420 0.8420 2.3 207 76 8.0 5 1 -0.2747 0.2747 3.6 208 119 6.3 5 2 0.9353 0.9353 2.5 209 70 6.4 1 0 -0.9203 0.9203 3.4 210 104 5.7 2 0 -0.4133 0.4133 2.6 211 38 9.9 2 5 0.5633 0.5633 5.6 212 70 6.0 4 2 0.1878 0.1878 3.3 213 88 6.5 4 1 0.0775 0.0775 3.0 214 30 5.8 5 4 -0.0271 0.0271 4.5 215 76 7.2 1 2 0.5797 0.5797 3.4 216 73 4.9 5 0 -1.2510 1.2510 3.1 217 54 8.3 5 5 0.5693 0.5693 4.4 218 28 5.1 3 4 0.1657 0.1657 4.3 219 64 6.7 4 2 0.0654 0.0654 3.6 220 28 5.0 1 3 0.1101 0.1101 4.3 221 37 6.4 1 1 -0.4992 0.4992 4.4 222 19 9.1 3 4 -0.0917 0.0917 6.3 223 59 7.8 3 1 -0.3572 0.3572 4.0 224 102 7.7 4 0 -0.7067 0.7067 2.9 225 34 9.6 3 2 -0.4264 0.4264 5.7 226 14 4.2 5 7 0.3781 0.3781 4.6 227 88 10.4 1 1 0.2973 0.2973 3.8 228 113 7.2 5 0 -0.6068 0.6068 2.7 229 72 5.8 4 1 -0.1857 0.1857 3.3 230 42 4.7 3 2 -0.1656 0.1656 3.8 231 18 9.3 2 4 -0.0083 0.0083 6.5 232 58 5.4 5 3 0.2061 0.2061 3.5 233 91 6.0 4 0 -0.8569 0.8569 2.9 234 67 8.9 4 1 -0.3485 0.3485 4.1 235 94 4.3 4 3 0.9678 0.9678 2.6 236 39 3.3 5 2 -0.3931 0.3931 3.6 237 79 8.2 4 1 -0.1209 0.1209 3.6 238 103 9.2 2 3 1.2171 1.2171 3.2 239 91 8.2 5 1 -0.0175 0.0175 3.3 240 101 9.4 4 2 0.6482 0.6482 3.3 241 59 7.6 2 0 -1.2473 1.2473 4.0 242 25 4.5 3 3 -0.1408 0.1408 4.3 243 11 7.4 3 7 0.4574 0.4574 5.9 244 21 8.8 3 6 0.4000 0.4000 6.1 245 40 5.9 5 1 -0.8528 0.8528 4.2 246 23 3.9 4 13 1.6070 1.6070 4.2 247 58 10.1 1 1 -0.2212 0.2212 4.7 248 24 9.0 1 3 -0.0585 0.0585 5.9 249 32 10.1 4 3 -0.2604 0.2604 6.0 250 118 4.1 2 2 1.2857 1.2857 2.2

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> summary(fm5)

Call:

lm(formula = sqrt(Y) ~ Dist + Inc + Size, weights = wi)

Residuals:

Min 1Q Median 3Q Max -3.1818 -0.8134 0.1118 0.9051 3.2484

Coefficients:

Dist -0.018063 0.001328 -13.599 < 2e-16 ***

Inc 0.027104 0.017669 1.534 0.126323#收入太低決定將其拿掉 Size 0.113891 0.029023 3.924 0.000113 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> anova(fm5)

Response: sqrt(Y)

Inc 1 2.14 2.14 1.4075 0.236623 Size 1 23.40 23.40 15.3988 0.000113 ***

Residuals 246 373.81 1.52 ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> par(mfrow=c(2,2))

> plot(fm5,ask=F)

(22)

> fm6=lm(sqrt(Y)~Dist+Size,weights=wi)

> summary(fm6)

Call:

lm(formula = sqrt(Y) ~ Dist + Size, weights = wi)

Residuals:

Min 1Q Median 3Q Max -3.2532 -0.8942 0.1028 0.9767 3.0845

Coefficients:

Dist -0.018099 0.001332 -13.590 < 2e-16 ***

Size 0.109893 0.028985 3.791 0.000188 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.236 on 247 degrees of freedom Multiple R-squared: 0.4458, Adjusted R-squared: 0.4413

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F-statistic: 99.34 on 2 and 247 DF, p-value: < 2.2e-16

> anova(fm6)

Response: sqrt(Y)

Size 1 21.96 21.96 14.374 0.0001884 ***

Residuals 247 377.38 1.53 ---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> par(mfrow=c(2,2))

> plot(fm6,ask=F)

最後我選擇 Y =2.077797 0.018099Dist− +0.109893Size這個 model, 但是其實還是有很多要改進，它的 R-squared 太低，解釋力不夠。或許利用 generalized least square 可以解決這個問題。