• Staphylococcus aureus (S. aureus) is ubiquitous and can be commensal or pathogenic.
  
• S. aureus has variable antibiotic resistance patterns, but resistance to methicillin is important.
  
• Methicillin is rarely used in treatment, but resistance to it is associated with resistance to other commonly used antibiotics.

• In particular S. aureus bacteremia (SAB) is common (3.6-6.0 cases per 100,000 py)
  - Community and hospital settings
  
• SAB in general has very high mortality and morbidity (~20-40% per episode)

AIM: Understand the relationship between methicillin-resistance and mortality in patients with SAB

• Serial enrollment of every patient with positive blood culture for S. aureus in SFGH lab from Jan 2008 - July 2013
• Database managed by infection control, but all patients had mandatory ID service consultation and recommendations
• Followed prospectively
• Outcome = death within 90 days (redundant: manual chart review, EMR query, online search, SSA MDF, CDC NDI database through 2011)

What is the causal effect of the presence versus absence of methicillin-resistance in the context of a first presentation of Staphylococcus aureus bloodstream infection on 90-day mortality among patients presenting to San Francisco General Hospital (SFGH)?

• Endogenous nodes: \(X = (W, A, Z, Y)\),
  where \(W\) is the vector of baseline covariates
• \(W_1\) is race (caucasian yes/no)
• \(W_2\) is sex
• \(W_3\) is age
• \(W_4\) is duration of bacteremia
• \(W_5\) is presumed source of bacteremia
• \(W_6\) is hospital onset of bacteremia
• \(W_7\) is homelessness
• \(W_8\) is IV drug use
• \(W_9\) is sepsis criteria on admit

• \(W_{10}\) prior hospitalization in the last year
• \(W_{11}\) is severity of comorbidities (by Charlson index)
• \(W_{12}\) is HIV
• \(W_{13}\) is cirrhosis
• \(W_{14}\) is is immunosuppresent use

\(A\) is absence or presence of methicillin-resistance
\(Z\) is a set of mediators denoting severity of illness presentation
- \(Z_1\) intensive care unit (ICU)
- \(Z_2\) length of hospital stay
\(Y\) is mortality

• Exogenous nodes: \(U = (U_W, U_A, U_Z, U_Y) \sim P_U\). There are no independence assumptions.
• Structural equations F:
• \(W = f_W (U_W)\)
• \(A = f_A(W, U_A)\)
• \(Z = f_Z(W, A, U_Z)\)
• \(Y = f_Y(W, A, Z, U_Y)\)
• Exclusion restrictions: We are assuming that the baseline covariates do not affect each other directly. We have not specified any functional forms.

• The observed data were generated by sampling n independent times from a data generating system described by the structural causal model MF. This yields n i.i.d. copies of random variable \(O = (W, A, Z, Y) \sim P_0\).
• The statistical model \(M\) for the set of allowed observed data distributions is non-parametric.

• Causal risk of death (within 90 days) due to the presence of methicillin-resistance in the context of a first presentation of Staphylococcus aureus bloodstream infection

\[\Psi_F (P_{U,X}) = P_{U,X}(Y_1 = 1) - PU,X(Y_0 = 1)\] \[\Psi_F= E_{U,X}(Y_1) - E_{U,X}(Y_0)\] where \(Y_a\) is the counterfactual outcome (90-day mortality), if the patient had methicillin-resistance A = a.

• We use \(M^{F*}\) to denote the original SCM augmented by the assumptions needed for identifiability. Under \(M^{F*}\), the backdoor criteria will hold conditionally on \(W\).

• We use \(M^{F*}\) to denote the original SCM augmented by the assumptions needed for identifiability. Under \(M^{F*}\), the backdoor criteria will hold conditionally on \(W\).

• For identifiability, we also need the positivity assumption to hold \(min_{a \in A} P_0(A=a|W=w) > 0\) for all \(w\) for which \(P_0(W = w) > 0\).
• There must be a positive probability of presence of methicillin-resistance or not, within in every covariate strata; W denotes the set of covariates that satisfy the backdoor criteria under the working \(M^{F*}\).

• The intervention to set A=a changes the value of Z and changes the value of Y. The effect of methicillin resistance on mortality has two pathways: directly (e.g., biological mechanism) and by changing severity of illness presentation. \[W = f_W(U_W)\] \[A = a\] \[Za = f_M(W, a, U_Z)\] \[Y = f_Y(W, a, Z_a, U_Y)\]

• For the analysis, ignore presence of Z. \[W = f_W(U_W)\] \[A = a\] \[Y = f_Y(W, a, U_Y)\]

• Under the working SCM \(M^{F*}\) and with the positivity assumption, the average treatment effect \(\Psi^F(P_U,X)\) is identified using the G-Computation formula: \[\Psi(P_0) = E_0[E_0(Y|A=1,W) - E_0(Y|A=0,W)]\]
  \[=\Sigma[E_0(Y|A=1,W=w)\] \[- E_0(Y|A=0,W=w)]P_0(W=w)\] where \(W\) is the vector of baseline covariates.

• Simple substitution estimator based on the G-Computation formula

\[\Psi_{SS}(P_n)= \frac{1}{n} \Sigma[\overline{\mbox{Q}}_n(1,Wi)-\overline{\mbox{Q}}_n(0,W_i)]\] where \(P_n\) is the empirical distribution and \(\overline{\mbox{Q}}_n(A,W)\) is the estimate of the conditional mean outcome given methicillin-resistance and baseline covariates \(E_0(Y|A,W)\).

• Inverse Probability of Treatment Weighting (IPTW) \[\Psi IPTW(P_n)= \frac{1}{n} \Sigma \frac{I(A_i=1)}{g_n(A_i|W_i)}Y_i\] \[ - \frac{1}{n} \Sigma\frac{I(A_i=0)}{g_n(A_i|W_i)} Y_i \]
  Where \(g_n(A_i|W_i)\) is an estimate of the conditional probability of methicillin-resistance given the baseline characteristics \(P_0(A|W)\). This conditional distribution is often referred to as the exposure or treatment mechanism.

• Targeted Maximum Likelihood Estimation

\[\Psi TMLE(P_n)= \frac{1}{n} \Sigma (\overline{\mbox{Q}}^*_n(1,W_i)-\overline{\mbox{Q}}^*_n(0,W_i)) \] Where \(\overline{\mbox{Q}}^*_n(A,W)\) is the updated estimate of the conditional mean outcome given methicillin-resistance and baseline covariates \(E_0(Y|A,W)\).

#SuperLearner model of Q(Y|A,W)
Qinit90D

## 
## Call:  
## SuperLearner(Y = ObsData$Y, X = X, newX = newdata, family = "binomial",  
##     SL.library = SL.library, method = "method.AUC", cvControl = list(V = 10L,  
##         stratifyCV = FALSE, shuffle = TRUE, validRows = NULL)) 
## 
## 
##                          Risk        Coef
## SL.glm_All          0.2599549  0.21119738
## SL.glm.EstA_All     0.3111445  0.11547734
## SL.glm.EstB_All     0.4250188 -0.08488417
## SL.glmnet_All       0.2896068  0.12512740
## SL.gam_All          0.2591034  0.25119178
## SL.polymars_All     0.3393188  0.20174807
## SL.randomForest_All 0.2822690  0.17152182

# pred prob of survival given A,W 
QbarAW <- Qinit90D$SL.predict[1:n]
Qbar1W <- Qinit90D$SL.predict[(n+1): (2*n)]
Qbar0W <- Qinit90D$SL.predict[(2*n+1): (3*n)]
PsiHat.SS<-mean(Qbar1W - Qbar0W) 
PsiHat.SS

## [1] 0.01131206

#Percent effect on 90 day mortality:
PsiHat.SS*100

## [1] 1.131206

#SuperLearner model of g(A|W)
gHatSL90D

## 
## Call:  
## SuperLearner(Y = ObsData$A, X = W, family = "binomial", SL.library = SL.library,  
##     method = "method.AUC", verbose = TRUE, cvControl = list(V = 10L,  
##         stratifyCV = FALSE, shuffle = TRUE, validRows = NULL)) 
## 
## 
##                          Risk      Coef
## SL.glm_All          0.4402636 0.1717388
## SL.glmnet_All       0.4830365 0.2041316
## SL.gam_All          0.4351526 0.2018086
## SL.polymars_All     0.4572965 0.2455639
## SL.randomForest_All 0.4461725 0.2184342

# generate the predicted prob of MRSA given baseline cov 
gHat1W<- gHatSL90D$SL.predict 
# generate the predicted prob of no MRSA, given baseline cov 
gHat0W<- 1- gHat1W
#IPTW Estimator:
PsiHat.IPTW1<- mean(as.numeric(ObsData$A==1)*ObsData$Y/gHat1W)
PsiHat.IPTW2<- mean(as.numeric(ObsData$A==0)*ObsData$Y/gHat0W) 
PsiHat.IPTW1 - PsiHat.IPTW2

## [1] -0.008696856

#Percent effect on 90 day mortality:
(PsiHat.IPTW1 - PsiHat.IPTW2)*100

## [1] -0.8696856

# 3. Estimate to create the clever covariate H_n^*(A,W)$ for each subject 
  H.AW<- ObsData$A/gHat1W - (1-ObsData$A)/gHat0W 
# also want to evaluate the clever covariates at A=1 and A=0  
  H.1W<- 1/gHat1W 
  H.0W<- -1/gHat0W
# 4. Update the initial estimate of Qbar_0(A,W) 
logitUpdate<- glm(ObsData$Y ~ -1 +offset(qlogis(QbarAW)) + H.AW, family=binomial)
eps<- logitUpdate$coef
# calc the predicted values for each subj under each txt 
QbarAW.star<- plogis(qlogis(QbarAW)+ eps*H.AW) 
Qbar1W.star<- plogis(qlogis(Qbar1W)+ eps*H.1W) 
Qbar0W.star<- plogis(qlogis(Qbar0W)+ eps*H.0W) 
# 5. Estimate Psi(P_0) as the emp mean of the difference in the pred 
# outcomes under A=1 and A=0
PsiHat.TMLE<- mean(Qbar1W.star - Qbar0W.star, na.rm=TRUE) 
PsiHat.TMLE

## [1] 0.01789815

#Percent effect on 90 day mortality:
PsiHat.TMLE*100

## [1] 1.789815

# evaluate the influence curve for all observations 
IC <- H.AW*(ObsData$Y - QbarAW.star) + Qbar1W.star - Qbar0W.star - PsiHat.TMLE 
summary(IC)

##        V1          
##  Min.   :-1.67606  
##  1st Qu.:-0.15624  
##  Median : 0.01898  
##  Mean   : 0.00000  
##  3rd Qu.: 0.13974  
##  Max.   : 1.82458

varHat.IC<-var(IC, na.rm=TRUE)/(n)
varHat.IC

##              [,1]
## [1,] 0.0005547578

# obtain confidence intervals
c(PsiHat.TMLE -1.96*sqrt(varHat.IC), PsiHat.TMLE +1.96*sqrt(varHat.IC))

## [1] -0.02826631  0.06406261

# calculate the pvalue 
2* pnorm( abs(PsiHat.TMLE / sqrt(varHat.IC)), lower.tail=F )

##           [,1]
## [1,] 0.4473144

• 400 Bootstrapped Samples

• Diagnosis with methicillin-resistant Staphylococcus aureus (MRSA) did not significatly affect mortality in the next 90 days among the population of patients at San Francisco General Hospital who were diagnosed with Staphylococcus aureus.
• Limitations
• Next steps?