# 第 63 章 混雜的調整，交互作用，和模型的可壓縮性

• 與關心的預測變量相關 (i.e. $$\delta_1 \neq 0$$)；
• 與因變量相關 (當關心的預測變量不變時，$$\beta_2\neq0$$ )；
• 不在預測變量和因變量的因果關係 (如果有的話) 中作媒介。Not be on the causal pathway between the predictor of interest and the dependent variable.

< 2cm Diameter
>= 2cm Diameter
Group Surgery Lithotripsy Surgery Lithotripsy
Success 81.00 234 192.00 55
Failure 6.00 36 71.00 25
Total 87.00 270 263.00 80
Odds Ratios 2.08 1.23

Outcome Surgery Lithotripsy
Success 273 (78%) 289 (83%)
Failure 77 61
Total 350 350
Odds ratio 0.75

Size of the Stone Surgery Lithotripsy
$$< 2$$ cm 87 (33%) 270 (77%)
$$\geqslant 2$$ cm 263 80
Total 350 350

Outcome $$< 2$$ cm $$\geqslant 2$$ cm
Success 234 (87%) 55 (69%)
Failure 36 25
Total 370 80

Outcome $$< 2$$ cm $$\geqslant 2$$ cm
Success 81 (93%) 192 (73%)
Failure 6 71
Total 87 263

## 63.1 混雜因素的調整

Group
$$X=0$$ $$X=1$$
$$D=0$$ $$X=0$$ $$n_{00}$$ $$n_{10}$$
$$X=1$$ $$n_{01}$$ $$n_{10}$$

### 63.1.1 Woolf 法估算合併比值比

$\text{Var}(\text{log}\hat\Psi_i) \approx \frac{1}{a_i} + \frac{1}{b_i} + \frac{1}{c_i} + \frac{1}{d_i} = \frac{1}{w_i}$

$\text{log}\hat\Psi_w = \frac{\sum w_i\text{log}\hat\Psi_i}{\sum w_i}$

$\text{Var}(\text{log}\hat\Psi_w) = \frac{1}{\sum w_i}$

\begin{aligned} \hat\Psi_1 = 2.08 ;&\; \hat\Psi_2 = 1.23 \\ \text{Var}(\text{log}\hat\Psi_1) = \frac{1}{81} & + \frac{1}{234} + \frac{1}{6} + \frac{1}{36} = 0.2111 \\ \text{Var}(\text{log}\hat\Psi_2) = \frac{1}{192} & + \frac{1}{55} + \frac{1}{71} + \frac{1}{25} = 0.0775 \\ w_1 = \frac{1}{\text{Var}(\text{log}\hat\Psi_1)} = 4.74 ; \;& w_2 = \frac{1}{\text{Var}(\text{log}\hat\Psi_2)} = 12.91 \\ \text{log}\hat\Psi_w = & \frac{4.74\times\text{log(2.08)} + 12.91\times\text{log(1.23)}}{4.74 + 12.91} \\ = & 0.3481 \\ \Rightarrow \hat\Psi_w =& e^{0.3481} = 1.42\\ \text{Var}(\hat\Psi_w) =& \frac{1}{4.74+12.91} = 0.0567 \\ \Rightarrow 95\% \text{ CI} = & e^{0.3481 \pm 1.96\times\sqrt{0.0567}} \\ = & (0.89, 2.26) \end{aligned}

Woolf 的計算調整後的合併比值比的方法是在線性迴歸和廣義線性迴歸被發現之前誕生的，但是其想法之精妙，確實令人感嘆。可惜其最大的缺陷是無法用這樣的方法進行連續型變量的調整，也很難同時進行多個變量的調整，所以現在這一算法已經逐漸被淘汰。現在我們有了廣義線性迴歸模型這一更強大的工具，只要把變量加入廣義線性模型進行調整就可以計算曾經難以計算和擴展的調整後的合併比值比。從下面的代碼計算獲得的調整後比值比 $$1.43 (0.91, 2.34)$$ 也可以看出，Woolf 方法的計算結果也是足夠令人滿意的。

size <- c("< 2cm", "< 2cm", ">= 2cm", ">= 2cm")
treatment <- c("Surgery","Lithotripsy","Surgery","Lithotripsy")
n <- c(87, 270, 263, 80)
Success <- c(81, 234, 192, 55)
Stone <- data.frame(size, treatment, n, Success)
ModelStone <- glm(cbind(Success, n - Success) ~ treatment + size,
family = binomial(link = logit), data = Stone)
summary(ModelStone)
##
## Call:
## glm(formula = cbind(Success, n - Success) ~ treatment + size,
##     family = binomial(link = logit), data = Stone)
##
## Deviance Residuals:
##        1         2         3         4
##  0.76357  -0.35881  -0.27563   0.46948
##
## Coefficients:
##                  Estimate Std. Error z value  Pr(>|z|)
## (Intercept)       1.93655    0.17045 11.3614 < 2.2e-16 ***
## treatmentSurgery  0.35723    0.22908  1.5594    0.1189
## size>= 2cm       -1.26057    0.23900 -5.2742 1.333e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 33.12395  on 3  degrees of freedom
## Residual deviance:  1.00816  on 1  degrees of freedom
## AIC: 26.3554
##
## Number of Fisher Scoring iterations: 3
broom::tidy(ModelStone, exp = FALSE, conf.int = TRUE) %>%
knitr::kable(.)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 1.93655073 0.17044973 11.3614184 0.00000000 1.61463718 2.28439560
treatmentSurgery 0.35722866 0.22907985 1.5594068 0.11890013 -0.09069343 0.80856158
size>= 2cm -1.26056536 0.23900383 -5.2742474 0.00000013 -1.73744172 -0.79909943

## 63.2 交互作用

ModelStone2 <- glm(cbind(Success, n - Success) ~ treatment*size,
family = binomial(link = logit), data = Stone)
summary(ModelStone2)
##
## Call:
## glm(formula = cbind(Success, n - Success) ~ treatment * size,
##     family = binomial(link = logit), data = Stone)
##
## Deviance Residuals:
## [1]  0  0  0  0
##
## Coefficients:
##                             Estimate Std. Error z value  Pr(>|z|)
## (Intercept)                  1.87180    0.17903 10.4553 < 2.2e-16 ***
## treatmentSurgery             0.73089    0.45942  1.5909 0.1116310
## size>= 2cm                  -1.08334    0.30039 -3.6065 0.0003104 ***
## treatmentSurgery:size>= 2cm -0.52453    0.53716 -0.9765 0.3288211
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 3.31239e+01  on 3  degrees of freedom
## Residual deviance: 1.39888e-14  on 0  degrees of freedom
## AIC: 27.3472
##
## Number of Fisher Scoring iterations: 3
broom::tidy(ModelStone2, exp = FALSE, conf.int = TRUE) %>%
knitr::kable(.)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 1.87180218 0.17902872 10.45531797 0.00000000 1.53494834 2.23884939
treatmentSurgery 0.73088751 0.45941624 1.59090482 0.11163100 -0.10165254 1.73013407
size>= 2cm -1.08334482 0.30038825 -3.60648201 0.00031038 -1.67042760 -0.48874515
treatmentSurgery:size>= 2cm -0.52452937 0.53715728 -0.97649124 0.32882109 -1.65338017 0.47642831

## 63.3 可壓縮性 collapsibility

### 63.3.1 線性迴歸的可壓縮性

$$Y$$ 標記結果變量，$$X$$ 標記暴露變量，$$Z$$ 則標記我們想要調整的某個混雜因子：

$Y = \alpha + \beta_X X + \beta_Z Z + \varepsilon, \text{ where } \varepsilon \sim N(0, \sigma^2)$

$E(Y | X) = \alpha + \beta_X X + \beta_Z E(Z|X)$

$E(Y|X) = \alpha + \beta_Z \mu_Z + \beta_X X$

### 63.3.2 邏輯鏈接方程時的不可壓縮性

Strata 1 Strata 2
Outcome Exposure $$+$$ Exposure $$-$$ Exposure $$+$$ Exposure $$-$$
Success 90 50 50 10
Failure 10 50 50 90
Total 100 100 100 100
Odds Ratios 9 9

Outcome Exposure $$+$$ Exposure $$-$$
Success 140 60
Failure 60 140
Total 200 200
Odds ratio 5.4

## 63.4 交互作用對尺度的依賴性

GLM 模型中的交互作用檢驗，對選用的尺度 (比值比 OR，還是危險度比 RR) 依賴性極高。用模型可壓縮性的數據例子也可以說明交互作用對尺度的依賴性。上文書說到，兩個分層中的比值比都是 9，該分層變量既沒有交互作用，也不是混淆因子 (當使用比值比的時候)。如果我們改用危險度比 (risk ratio, RR)，在分類變量的第一層 (Strata 1) 中，暴露的危險度比是 $$\frac{90/100}{50/100} = 1.8$$；分類變量的第二層 (Strata 2) 中，暴露的危險度比是 $$\frac{50/100}{10/100} = 5$$。所以使用危險度比作爲評價指標的時候，被調整的分類變量就突然搖身一變變成了有交互作用的因子。這裏，我們用數據，證明了交互作用的存在與否，對尺度的選用依賴性極高。這就導致我們在描述一個變量是否對我們關心的暴露和結果之間的關係有交互作用時，必須明確指出所使用的是比值比，還是危險度比進行的交互作用評價。

## 63.5 GLM Practical 07

• treat 表示患者是接收了新療法 (1 = new)，或者繼續維持現有的療法 (0 = current)；

• basecontrol 表示患者在剛進入實驗時 (baseline) 血壓原本控制的狀態 (0 = bad; 1 = good)；

• fupcontrol 表示患者在實驗過程的隨訪結果中 (followup) 血壓原本控制的狀態 (0 = bad; 1 = good)；

### 63.5.1 使用你熟悉的統計學軟件擬合一個由 fupcontrol 作爲結果變量，treat 作爲唯一預測變量的廣義線性回歸模型。 根據報告的結果，寫一段適用於醫學/流行病學文獻雜誌的報告。

highbp <- read_dta("../backupfiles/highbp.dta")

m0 <- glm(fupcontrol ~ treat, data = highbp,
summary(m0); jtools::summ(m0, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = fupcontrol ~ treat, family = binomial(link = logit),
##     data = highbp)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -1.18217  -0.85608  -0.85608   1.17266   1.53724
##
## Coefficients:
##              Estimate Std. Error z value  Pr(>|z|)
## (Intercept) -0.815122   0.043368 -18.795 < 2.2e-16 ***
## treat        0.826323   0.058999  14.006 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6749.10  on 4999  degrees of freedom
## Residual deviance: 6548.24  on 4998  degrees of freedom
## AIC: 6552.24
##
## Number of Fisher Scoring iterations: 4
 Observations 5000 Dependent variable fupcontrol Type Generalized linear model Family binomial Link logit
 𝛘²(1) 200.861 Pseudo-R² (Cragg-Uhler) 0.0531595 Pseudo-R² (McFadden) 0.0297611 AIC 6552.24 BIC 6565.27
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 0.4425851 0.4065197 0.4818502 -18.7953325 0.0000000
treat 2.2849008 2.0353891 2.5649993 14.0057422 0.0000000
Standard errors: MLE

There was strong evidence (p < 0.001) that treatment is associated with the probability of having BP controlled at followup, with patients randomised to the treatment having odds of BP controlled of 2.28 (95%CI 2.04 to 2.56) higher than those randomised to the current treatment.

### 63.5.2 分析 treat 和 basecontrol 之間的關係，結果是否如你的預期那樣？

m1 <- glm(basecontrol ~ treat, data = highbp,
summary(m1); jtools::summ(m1, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = basecontrol ~ treat, family = binomial(link = logit),
##     data = highbp)
##
## Deviance Residuals:
##     Min       1Q   Median       3Q      Max
## -1.1972  -1.1849   1.1578   1.1699   1.1699
##
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.017600   0.040002  0.4400   0.6599
## treat       0.028808   0.056577  0.5092   0.6106
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6930.19  on 4999  degrees of freedom
## Residual deviance: 6929.93  on 4998  degrees of freedom
## AIC: 6933.93
##
## Number of Fisher Scoring iterations: 3
 Observations 5000 Dependent variable basecontrol Type Generalized linear model Family binomial Link logit
 𝛘²(1) 0.259269 Pseudo-R² (Cragg-Uhler) 6.91e-05 Pseudo-R² (McFadden) 3.74e-05 AIC 6933.93 BIC 6946.97
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 1.0177563 0.9410103 1.1007613 0.4399943 0.6599412
treat 1.0292268 0.9211969 1.1499256 0.5091777 0.6106277
Standard errors: MLE

### 63.5.3 已知模型中如果增加調整基線變量可能對 fupcontrol 有一定的預測效果。 在你的模型中增加基線血壓控制情況的變量。與 m0 的結果 (治療效果 treatment effect；參數標準誤 standard error；和 p 值)。重新修改之前用於發表在醫學雜誌上關於這個分析結果的報告描述。

m2 <- glm(fupcontrol ~ treat + basecontrol, data = highbp,
summary(m2); jtools::summ(m2, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = fupcontrol ~ treat + basecontrol, family = binomial(link = logit),
##     data = highbp)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -1.60531  -0.79467  -0.50514   0.80315   2.06015
##
## Coefficients:
##              Estimate Std. Error z value  Pr(>|z|)
## (Intercept) -1.994537   0.067478 -29.558 < 2.2e-16 ***
## treat        1.003756   0.066550  15.083 < 2.2e-16 ***
## basecontrol  1.956770   0.067984  28.783 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6749.10  on 4999  degrees of freedom
## Residual deviance: 5588.19  on 4997  degrees of freedom
## AIC: 5594.19
##
## Number of Fisher Scoring iterations: 4
 Observations 5000 Dependent variable fupcontrol Type Generalized linear model Family binomial Link logit
 𝛘²(2) 1160.91 Pseudo-R² (Cragg-Uhler) 0.279729 Pseudo-R² (McFadden) 0.17201 AIC 5594.19 BIC 5613.74
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 0.1360766 0.1192192 0.1553176 -29.5584135 0.0000000
treat 2.7285122 2.3948517 3.1086597 15.0827896 0.0000000
basecontrol 7.0764349 6.1936470 8.0850477 28.7828138 0.0000000
Standard errors: MLE

After adjusting for baseline BP control, the estimated log OR for the new vs current treatment is 1.00 (95%CI 0.87 to 1.13). This is quite a bit larger than the corresponding estimate from part 1, which was 0.83 (0.71, 0.94). The standard error has, perhaps contrary to expectation, increased from 0.059 to 0.067. The p-values are highly significant from both analyses, but the z-statistics is larger in the baseline adjusted analysis, which indicates this result is more statistically significant.

The increase in the log OR estimate is due to the fact that the baseline adjusted analysis is estimating a different parameter. This is because, although there is no confounding, odds ratios are not collapsible - a conditional odds ratio has a different interpretation from a marginal one. The baseline adjusted analyses estimates the odds ratio for two patients who have the same value of baseline BP control, and this differs from the unconditional OR.

### 63.5.5 實驗研究者更想知道新的治療方案是否由於基線時患者的血壓控制情況而有不同。爲了回答這個問題，請擬合對應的廣義線性回歸模型。根據結果回答這個問題。

m3 <- glm(fupcontrol ~ treat + basecontrol + treat*basecontrol,
data = highbp, family = binomial(link = logit))
summary(m3); jtools::summ(m3, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = fupcontrol ~ treat + basecontrol + treat * basecontrol,
##     family = binomial(link = logit), data = highbp)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -1.59756  -0.80088  -0.49709   0.80906   2.07473
##
## Coefficients:
##                    Estimate Std. Error  z value Pr(>|z|)
## (Intercept)       -2.028696   0.088642 -22.8863   <2e-16 ***
## treat              1.056110   0.109413   9.6525   <2e-16 ***
## basecontrol        2.004905   0.105024  19.0900   <2e-16 ***
## treat:basecontrol -0.083509   0.137947  -0.6054   0.5449
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6749.10  on 4999  degrees of freedom
## Residual deviance: 5587.82  on 4996  degrees of freedom
## AIC: 5595.82
##
## Number of Fisher Scoring iterations: 4
 Observations 5000 Dependent variable fupcontrol Type Generalized linear model Family binomial Link logit
 𝛘²(3) 1161.28 Pseudo-R² (Cragg-Uhler) 0.279807 Pseudo-R² (McFadden) 0.172065 AIC 5595.82 BIC 5621.89
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 0.1315068 0.1105340 0.1564591 -22.8862857 0.0000000
treat 2.8751646 2.3202252 3.5628315 9.6525024 0.0000000
basecontrol 7.4253853 6.0439727 9.1225341 19.0899856 0.0000000
treat:basecontrol 0.9198831 0.7019588 1.2054622 -0.6053666 0.5449354
Standard errors: MLE
lrtest(m3, m2)
## Likelihood ratio test
##
## Model 1: fupcontrol ~ treat + basecontrol + treat * basecontrol
## Model 2: fupcontrol ~ treat + basecontrol
##   #Df   LogLik Df   Chisq Pr(>Chisq)
## 1   4 -2793.91
## 2   3 -2794.09 -1 0.36737    0.54444

### 63.5.6 換一個模型，先不考慮 basecontrol，使用危險度比 (risk ratio) 來評價不同治療方案之間的療效。

m4 <- glm(fupcontrol ~ treat,
data = highbp, family = binomial(link = log))
summary(m4); jtools::summ(m4, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = fupcontrol ~ treat, family = binomial(link = log),
##     data = highbp)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -1.18217  -0.85608  -0.85608   1.17266   1.53724
##
## Coefficients:
##              Estimate Std. Error z value  Pr(>|z|)
## (Intercept) -1.181559   0.030058 -39.310 < 2.2e-16 ***
## treat        0.493996   0.036042  13.706 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6749.10  on 4999  degrees of freedom
## Residual deviance: 6548.24  on 4998  degrees of freedom
## AIC: 6552.24
##
## Number of Fisher Scoring iterations: 5
 Observations 5000 Dependent variable fupcontrol Type Generalized linear model Family binomial Link log
 𝛘²(1) 200.861 Pseudo-R² (Cragg-Uhler) 0.0531595 Pseudo-R² (McFadden) 0.0297611 AIC 6552.24 BIC 6565.27
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 0.3068000 0.2892478 0.3254173 -39.3095666 0.0000000
treat 1.6388526 1.5270777 1.7588090 13.7062367 0.0000000
Standard errors: MLE

### 63.5.7 在前一模型m4中加入 basecontrol，與未加入該變量時模型的輸出結果相比，有什麼不同？

m5 <- glm(fupcontrol ~ treat + basecontrol,
data = highbp, family = binomial(link = log))
summary(m5); jtools::summ(m5, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = fupcontrol ~ treat + basecontrol, family = binomial(link = log),
##     data = highbp)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -1.62393  -0.74052  -0.57858   0.78906   1.93392
##
## Coefficients:
##              Estimate Std. Error z value  Pr(>|z|)
## (Intercept) -1.870031   0.045140 -41.427 < 2.2e-16 ***
## treat        0.442106   0.031752  13.924 < 2.2e-16 ***
## basecontrol  1.116613   0.043236  25.826 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6749.10  on 4999  degrees of freedom
## Residual deviance: 5613.89  on 4997  degrees of freedom
## AIC: 5619.89
##
## Number of Fisher Scoring iterations: 6
 Observations 5000 Dependent variable fupcontrol Type Generalized linear model Family binomial Link log
 𝛘²(2) 1135.21 Pseudo-R² (Cragg-Uhler) 0.274212 Pseudo-R² (McFadden) 0.168202 AIC 5619.89 BIC 5639.44
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 0.1541189 0.1410694 0.1683756 -41.4273054 0.0000000
treat 1.5559807 1.4620984 1.6558911 13.9236039 0.0000000
basecontrol 3.0544926 2.8063141 3.3246190 25.8258341 0.0000000
Standard errors: MLE

### 63.5.8 給上述模型增加交互作用項。對於危險度比作爲指標時的交互作用分析結果，和使用比值比時相比，你有怎樣的思考和結論？

m6 <- glm(fupcontrol ~ treat + basecontrol + treat*basecontrol,
data = highbp, family = binomial(link = log))
summary(m6); jtools::summ(m6, exp = TRUE, confint = TRUE, digits = 7)
##
## Call:
## glm(formula = fupcontrol ~ treat + basecontrol + treat * basecontrol,
##     family = binomial(link = log), data = highbp)
##
## Deviance Residuals:
##      Min        1Q    Median        3Q       Max
## -1.59756  -0.80088  -0.49709   0.80906   2.07473
##
## Coefficients:
##                    Estimate Std. Error  z value  Pr(>|z|)
## (Intercept)       -2.152247   0.078341 -27.4728 < 2.2e-16 ***
## treat              0.858952   0.091123   9.4263 < 2.2e-16 ***
## basecontrol        1.447133   0.083363  17.3594 < 2.2e-16 ***
## treat:basecontrol -0.481126   0.097048  -4.9576 7.136e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6749.10  on 4999  degrees of freedom
## Residual deviance: 5587.82  on 4996  degrees of freedom
## AIC: 5595.82
##
## Number of Fisher Scoring iterations: 6
 Observations 5000 Dependent variable fupcontrol Type Generalized linear model Family binomial Link log
 𝛘²(3) 1161.28 Pseudo-R² (Cragg-Uhler) 0.279807 Pseudo-R² (McFadden) 0.172065 AIC 5595.82 BIC 5621.89
exp(Est.) 2.5% 97.5% z val. p
(Intercept) 0.1162228 0.0996798 0.1355112 -27.4728008 0.0000000
treat 2.3606845 1.9745766 2.8222919 9.4262970 0.0000000
basecontrol 4.2509087 3.6101303 5.0054218 17.3593671 0.0000000
treat:basecontrol 0.6180868 0.5110253 0.7475780 -4.9576328 0.0000007
Standard errors: MLE
lrtest(m6, m5)
## Likelihood ratio test
##
## Model 1: fupcontrol ~ treat + basecontrol + treat * basecontrol
## Model 2: fupcontrol ~ treat + basecontrol
##   #Df   LogLik Df   Chisq Pr(>Chisq)
## 1   4 -2793.91
## 2   3 -2806.95 -1 26.0728 3.2878e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

### 63.5.9 如果說不考慮一個RCT的統計分析不能在收集完數據之後修改這一事實，你認爲危險度比模型和比值比模型更應該使用哪一個來總結本數據的結果呢？

Since there is no interaction on the log odds scale, using a logisitic regression is probably preferable, as it gives a simpler model which correctly models the data. Whether the unadjusted or baseline adjusted results are presented is a question which is still hotly debated. The latter has increased power to detect a treatment effect, but as we have seen it estimates a different parameter to the marginal unadjusted OR.