Parameter approaches to estimate the Force of Infection
Enwu Liu
2023-06-03
Notes to estimate the Force of Infection using linear regression models.
glm(cbind(Pos,Tot-Pos), we use cbind() here, because the data are not individual observations, it is a kind of weighted data.
X<–matrix(rep(1,length(Age))), x was assigned to negative because FOI need to be positive and for the catalytic model we need to restrict coefficient to be negative, since disease prevalence need to between 0 to 1.
The “5” in the code is used to adjust the scale of sero-prevalece axis and the FOI axis, since the two axises have different scales.
hav<-read.table("C:/Users/enliu/OneDrive - Australian Catholic University/Modelling infectious diseases/Chapter4/HAV-BUL.dat",header=T)
hav## Age Pos Tot
## 1 1 3 16
## 2 2 3 15
## 3 3 3 16
## 4 4 4 13
## 5 5 7 12
## 6 6 4 15
## 7 7 3 12
## 8 8 4 11
## 9 9 7 10
## 10 10 8 15
## 11 11 2 7
## 12 12 3 7
## 13 13 2 11
## 14 14 0 1
## 15 15 5 16
## 16 16 13 41
## 17 17 1 2
## 18 18 3 6
## 19 19 15 32
## 20 20 22 37
## 21 21 15 24
## 22 22 7 10
## 23 23 8 10
## 24 24 7 11
## 25 25 12 15
## 26 26 5 10
## 27 27 10 13
## 28 28 15 19
## 29 29 9 12
## 30 30 9 9
## 31 31 9 14
## 32 32 8 10
## 33 33 9 11
## 34 34 8 9
## 35 35 9 14
## 36 36 13 14
## 37 37 6 7
## 38 38 15 16
## 39 39 11 13
## 40 40 6 8
## 41 41 8 8
## 42 42 13 14
## 43 43 7 10
## 44 44 5 5
## 45 45 7 7
## 46 46 9 9
## 47 47 9 9
## 48 48 22 22
## 49 49 6 7
## 50 50 10 10
## 51 51 6 6
## 52 52 13 14
## 53 53 8 8
## 54 54 7 7
## 55 55 13 13
## 56 56 11 11
## 57 57 8 8
## 58 58 8 8
## 59 59 9 10
## 60 60 13 16
## 61 61 5 5
## 62 62 5 6
## 63 63 5 5
## 64 64 5 5
## 65 65 10 10
## 66 66 8 8
## 67 67 4 4
## 68 68 5 5
## 69 69 4 5
## 70 70 8 8
## 71 72 9 9
## 72 73 1 1
## 73 74 4 4
## 74 75 7 7
## 75 76 6 6
## 76 77 2 2
## 77 78 3 3
## 78 79 2 2
## 79 80 4 4
## 80 81 1 1
## 81 82 1 1
## 82 83 2 2
## 83 86 1 1attach(hav)
### Muench's Model ###
#######################
model1<-glm(cbind(Tot-Pos,Pos)~-1+Age,family=binomial(link="log"))
summary(model1)##
## Call:
## glm(formula = cbind(Tot - Pos, Pos) ~ -1 + Age, family = binomial(link = "log"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.6876 -0.9677 -0.3434 0.2554 3.0612
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## Age -0.050500 0.002457 -20.55 <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: Inf on 83 degrees of freedom
## Residual deviance: 97.275 on 82 degrees of freedom
## AIC: 219.19
##
## Number of Fisher Scoring iterations: 5# Alternative R-code to fit Muench's model
# model2<-glm(cbind(Pos,Tot-Pos)~1,offset=log(Age),
# family=binomial(link="cloglog"))
# summary(model2)
# exp(coef(model2))
### Griffiths' Model ###
#########################
model3<-glm(cbind(Tot-Pos,Pos)~-1+Age+I(Age^2),
family=binomial(link="log"))
summary(model3)##
## Call:
## glm(formula = cbind(Tot - Pos, Pos) ~ -1 + Age + I(Age^2), family = binomial(link = "log"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.8864 -0.8813 -0.3290 0.1780 2.7599
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## Age -0.0442616 0.0053697 -8.243 <2e-16 ***
## I(Age^2) -0.0001889 0.0001491 -1.266 0.205
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: Inf on 83 degrees of freedom
## Residual deviance: 95.445 on 81 degrees of freedom
## AIC: 219.36
##
## Number of Fisher Scoring iterations: 6### Grenfell and Anderson ###
#############################
model4<-glm(cbind(Tot-Pos,Pos)~-1+Age+I(Age^2)+I(Age^3),
family=binomial(link="log"))
summary(model4)##
## Call:
## glm(formula = cbind(Tot - Pos, Pos) ~ -1 + Age + I(Age^2) + I(Age^3),
## family = binomial(link = "log"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.6704 -0.9169 -0.2430 0.1743 2.8057
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## Age -5.326e-02 1.032e-02 -5.159 2.49e-07 ***
## I(Age^2) 5.065e-04 6.508e-04 0.778 0.436
## I(Age^3) -1.019e-05 9.149e-06 -1.114 0.265
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: Inf on 83 degrees of freedom
## Residual deviance: 93.797 on 80 degrees of freedom
## AIC: 219.72
##
## Number of Fisher Scoring iterations: 8### FIGURE 6.1
#windows(record=TRUE, width=5, height=5)
par(las=1,cex.axis=1.1,cex.lab=1.1,lwd=3,mgp=c(2, 0.5, 0),mar=c(4.1,4.1,4.1,3))
plot(Age,Pos/Tot,cex=0.1*Tot,xlab="age",xlim=c(0,86),ylim=c(0,1),ylab="seroprevalence")
lines(Age,1-model1$fitted.values,lwd=2,col='red')
lines(Age,1-model3$fitted.values,lwd=2,lty=2,col='blue')
lines(Age,1-model4$fitted.values,lwd=2,lty=3,col='orange')
X<--matrix(rep(1,length(Age)))
lines(Age,5*X%*%model1$coefficients,lwd=2) #note there are two axis, the sero-prevalence and FOI are at different scale
X<--cbind(rep(1,length(Age)),2*Age)
lines(Age,5*X%*%model3$coefficients,lwd=2,lty=2)
X<--cbind(rep(1,length(Age)),2*Age,3*Age^2)
lines(Age,5*X%*%model4$coefficients,lwd=2,lty=3)
axis(side=4,at=c(0.0,0.2,0.4),labels=c(0.00,0.04,0.08))
mtext(side=4,"force of infection", las=3,line=2)
R code using polynomial regression to estimate the force of infection for Hepatitis A. We need to pay attention to the following equations.
For model 1 with the logit link, the force of infection was calculated as \[\lambda(a)=\eta'(a)\frac{e^{\eta(a)}}{1+e^{\eta(a)}}\] (red solid line)
For model 1 with logit link, \(\eta(a)=\beta_0+\beta_1*Age+\beta_2*Age^3\) therefore, \(\eta'(a)=\beta_1+3\beta_2 Age^2\) and \(\delta(\eta(a))=\frac{e^{\eta(a)}}{1+e^{\eta(a)}}\) are the fitted values from the regression model 1.
For model 2 with the complementary log-log link, the force of infection was calculated as \(\lambda(a)=\eta'(a)e^{\eta(a)}\)
For model 2, \(\eta(a)=\beta_0+\beta_1*Age^2+\beta_2*Age^3\) therefore, \[\eta'(a)=2\beta_1+3\beta_2Age^2\] and \(\delta(\eta(a))=e^{(\beta_0+\beta_1*Age^2+\beta_2*Age^3}\). Therefore,
\[\lambda(a)=(2\beta_1+3\times\beta_2Age^2)\times e^{(\beta_0+\beta_1*Age^2+\beta_2*Age^3)}\] (blue dashed line)
### DATA HAV-BE
hav<-read.table("C:/Users/enliu/OneDrive - Australian Catholic University/Modelling infectious diseases/Chapter4/HAV-BE.dat",header=T)
attach(hav)
hav## Id Age Pos Neg Tot
## 1 1 0.5 4 0 4
## 2 2 1.0 11 0 11
## 3 3 2.0 6 0 6
## 4 4 3.0 20 2 22
## 5 5 4.0 28 2 30
## 6 6 5.0 55 1 56
## 7 7 6.0 70 3 73
## 8 8 7.0 38 5 43
## 9 9 8.0 37 2 39
## 10 10 9.0 31 3 34
## 11 11 10.0 35 1 36
## 12 12 11.0 27 1 28
## 13 13 12.0 29 1 30
## 14 14 13.0 25 2 27
## 15 15 14.0 25 2 27
## 16 16 15.0 26 1 27
## 17 17 16.0 16 2 18
## 18 18 17.0 18 5 23
## 19 19 18.0 16 4 20
## 20 20 19.0 16 0 16
## 21 21 20.0 31 4 35
## 22 22 21.0 34 10 44
## 23 23 22.0 31 11 42
## 24 24 23.0 36 9 45
## 25 25 24.0 35 9 44
## 26 26 25.0 25 8 33
## 27 27 26.0 36 10 46
## 28 28 27.0 46 19 65
## 29 29 28.0 36 8 44
## 30 30 29.0 31 12 43
## 31 31 30.0 30 10 40
## 32 32 31.0 22 20 42
## 33 33 32.0 27 15 42
## 34 34 33.0 29 20 49
## 35 35 34.0 30 23 53
## 36 36 35.0 29 23 52
## 37 37 36.0 23 32 55
## 38 38 37.0 17 29 46
## 39 39 38.0 20 24 44
## 40 40 39.0 20 29 49
## 41 41 40.0 20 37 57
## 42 42 41.0 22 26 48
## 43 43 42.0 26 38 64
## 44 44 43.0 18 42 60
## 45 45 44.0 13 42 55
## 46 46 45.0 18 37 55
## 47 47 46.0 21 34 55
## 48 48 47.0 21 52 73
## 49 49 48.0 18 39 57
## 50 50 49.0 20 44 64
## 51 51 50.0 18 44 62
## 52 52 51.0 13 38 51
## 53 53 52.0 12 32 44
## 54 54 53.0 8 47 55
## 55 55 54.0 7 64 71
## 56 56 55.0 14 57 71
## 57 57 56.0 7 64 71
## 58 58 57.0 11 62 73
## 59 59 58.0 8 40 48
## 60 60 59.0 8 47 55
## 61 61 60.0 8 33 41
## 62 62 61.0 4 26 30
## 63 63 62.0 4 16 20
## 64 64 63.0 6 23 29
## 65 65 64.0 4 20 24
## 66 66 65.0 4 24 28
## 67 67 66.0 7 15 22
## 68 68 67.0 4 24 28
## 69 69 68.0 2 22 24
## 70 70 69.0 3 17 20
## 71 71 70.0 3 19 22
## 72 72 71.0 4 20 24
## 73 73 72.0 5 7 12
## 74 74 73.0 2 18 20
## 75 75 74.0 3 12 15
## 76 76 75.0 3 10 13
## 77 77 76.0 0 15 15
## 78 78 77.0 1 12 13
## 79 79 78.0 4 9 13
## 80 80 79.0 2 14 16
## 81 81 80.0 2 10 12
## 82 82 81.0 4 12 16
## 83 83 82.0 1 9 10
## 84 84 83.0 1 6 7
## 85 85 84.0 1 8 9
## 86 86 85.0 1 5 6names(hav)## [1] "Id" "Age" "Pos" "Neg" "Tot"pos<-hav$Neg
neg<-hav$Pos
Tot<-hav$Tot
model.fp1<-glm(cbind(pos,Tot-pos) ~ Age + I(Age^3), family=binomial(link="logit")) #n(a)=beta0+beta1*a+beta2*a^3
summary(model.fp1)##
## Call:
## glm(formula = cbind(pos, Tot - pos) ~ Age + I(Age^3), family = binomial(link = "logit"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.32807 -0.68338 -0.08881 0.61747 2.45576
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.069e+00 2.008e-01 -20.269 <2e-16 ***
## Age 1.241e-01 6.743e-03 18.406 <2e-16 ***
## I(Age^3) -8.198e-06 9.545e-07 -8.589 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1320.45 on 85 degrees of freedom
## Residual deviance: 82.74 on 83 degrees of freedom
## AIC: 362.33
##
## Number of Fisher Scoring iterations: 4model.fp2<-glm(cbind(pos,Tot-pos) ~ I(Age^2) + I(Age^3), family=binomial(link="cloglog")) #n(a)=beta0+beta1*a^2+beta2*a^3
summary(model.fp2)##
## Call:
## glm(formula = cbind(pos, Tot - pos) ~ I(Age^2) + I(Age^3), family = binomial(link = "cloglog"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.7053 -0.8777 -0.2868 0.6961 2.3852
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.355e+00 9.173e-02 -25.67 <2e-16 ***
## I(Age^2) 2.189e-03 9.985e-05 21.92 <2e-16 ***
## I(Age^3) -2.216e-05 1.208e-06 -18.35 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1320.45 on 85 degrees of freedom
## Residual deviance: 119.92 on 83 degrees of freedom
## AIC: 399.51
##
## Number of Fisher Scoring iterations: 5### FIGURE 6.4
#windows(record=TRUE, width=5, height=5)
par(las=1,cex.axis=1.1,cex.lab=1.1,lwd=3,mgp=c(2, 0.5, 0),mar=c(4.1,4.1,4.1,3))
plot(Age,pos/Tot,cex=0.05*Tot,xlab="age",ylab="seroprevalence",xlim=c(0,86),ylim=c(-0.2,1),yaxt="n")
lines(Age,model.fp1$fitted.values,lwd=2,col='red')
lines(Age,model.fp2$fitted.values,lwd=2,lty=2,col='blue')
model.fp1.foi<-(model.fp1$coef[2]+3*model.fp1$coef[3]*Age^2)*model.fp1$fitted.values #logit link n'(a)*fitted values
model.fp2.foi<-(2*model.fp2$coef[2]*Age+3*model.fp2$coef[3]*Age^2)*exp(model.fp2$coef[1]+model.fp2$coef[2]*Age^2+model.fp2$coef[3]*Age^3) #complementary log-log link n'(a)*e^{n(a)}
lines(Age,4*model.fp1.foi,lwd=2,col='red') # 4 is for axis scale
lines(Age,4*model.fp2.foi,lwd=2,lty=2,col='blue')
abline(0,0,lwd=2)
axis(side=4,at=c(-0.2,0.0,0.2,0.4),labels=c(-0.2,0.0,0.2,0.4))
axis(side=2,at= seq(0,1,0.2),labels=seq(0,1,0.2))
mtext(side=4,"force of infection", las=3,line=2)
From the above figure, we see that force of infection becomes negative at older age, to overcome the negative foi, a fractional polynomial regression with monotone restriction on the prevalence can be used, i.e we can force the sero-prevalence increasing with age.
hav<-read.table("C:/Users/enliu/OneDrive - Australian Catholic University/Modelling infectious diseases/Chapter4/HAV-BUL.dat",header=T)
hav## Age Pos Tot
## 1 1 3 16
## 2 2 3 15
## 3 3 3 16
## 4 4 4 13
## 5 5 7 12
## 6 6 4 15
## 7 7 3 12
## 8 8 4 11
## 9 9 7 10
## 10 10 8 15
## 11 11 2 7
## 12 12 3 7
## 13 13 2 11
## 14 14 0 1
## 15 15 5 16
## 16 16 13 41
## 17 17 1 2
## 18 18 3 6
## 19 19 15 32
## 20 20 22 37
## 21 21 15 24
## 22 22 7 10
## 23 23 8 10
## 24 24 7 11
## 25 25 12 15
## 26 26 5 10
## 27 27 10 13
## 28 28 15 19
## 29 29 9 12
## 30 30 9 9
## 31 31 9 14
## 32 32 8 10
## 33 33 9 11
## 34 34 8 9
## 35 35 9 14
## 36 36 13 14
## 37 37 6 7
## 38 38 15 16
## 39 39 11 13
## 40 40 6 8
## 41 41 8 8
## 42 42 13 14
## 43 43 7 10
## 44 44 5 5
## 45 45 7 7
## 46 46 9 9
## 47 47 9 9
## 48 48 22 22
## 49 49 6 7
## 50 50 10 10
## 51 51 6 6
## 52 52 13 14
## 53 53 8 8
## 54 54 7 7
## 55 55 13 13
## 56 56 11 11
## 57 57 8 8
## 58 58 8 8
## 59 59 9 10
## 60 60 13 16
## 61 61 5 5
## 62 62 5 6
## 63 63 5 5
## 64 64 5 5
## 65 65 10 10
## 66 66 8 8
## 67 67 4 4
## 68 68 5 5
## 69 69 4 5
## 70 70 8 8
## 71 72 9 9
## 72 73 1 1
## 73 74 4 4
## 74 75 7 7
## 75 76 6 6
## 76 77 2 2
## 77 78 3 3
## 78 79 2 2
## 79 80 4 4
## 80 81 1 1
## 81 82 1 1
## 82 83 2 2
## 83 86 1 1names(hav)## [1] "Age" "Pos" "Tot"search.fracpoly.twoR<-function(y,tot,x,pow,mc){
deviance<-deviance(glm(cbind(y,tot-y)~x+I(x^2), family="binomial"(link=logit)))
mistake<-NULL
for (i in 1: (length(pow))){
for (j in i: (length(pow))){
if(pow[i]==0){term1<-log(x)} else{term1<-(x)^(pow[i])}
if(pow[j]==pow[i]){term2<-term1*log(x)}
else if(pow[j]==0){term2<-log(x)}
else{term2<-(x)^(pow[j])}
glm.try<-glm(cbind(y,tot-y)~term1+term2, family="binomial"(link=logit))
print(c(pow[i],pow[j],deviance(glm.try),(sum(diff(predict(glm.try))<0)==0)))
if(glm.try$converged==FALSE){mistake<-rbind(mistake, c(1,pow[i],pow[j]))}
else{
if(deviance(glm.try)<deviance){
if (((mc)&&(sum(diff(predict(glm.try))<0)==0))|(!mc)){
#print(c(mc,(sum(diff(predict(glm.try))<0)==0)))
deviance<-deviance(glm.try)
power<-c(pow[i],pow[j])
}
}
}
}
}
return(list(power=power, deviance=deviance, mistake=mistake))
}
search.fracpoly.twoR(y=hav$Pos,tot=hav$Tot,x=hav$Age,seq(-2,3,0.1),T)## [1] -2.0000 -2.0000 244.5834 0.0000
## [1] -2.0000 -1.9000 241.1363 0.0000
## [1] -2.0000 -1.8000 237.3267 0.0000
## [1] -2.0000 -1.7000 233.1234 0.0000
## [1] -2.0000 -1.6000 228.4969 0.0000
## [1] -2.0000 -1.5000 223.4211 0.0000
## [1] -2.0000 -1.4000 217.8759 0.0000
## [1] -2.0000 -1.3000 211.8495 0.0000
## [1] -2.0000 -1.2000 205.3414 0.0000
## [1] -2.0000 -1.1000 198.3659 0.0000
## [1] -2.0000 -1.0000 190.9538 0.0000
## [1] -2.0000 -0.9000 183.1551 0.0000
## [1] -2.0000 -0.8000 175.0387 0.0000
## [1] -2.0000 -0.7000 166.6924 0.0000
## [1] -2.0000 -0.6000 158.2196 0.0000
## [1] -2.0000 -0.5000 149.7355 0.0000
## [1] -2.0000 -0.4000 141.3617 0.0000
## [1] -2.0000 -0.3000 133.2201 0.0000
## [1] -2.0000 -0.2000 125.4266 0.0000
## [1] -2.0000 -0.1000 118.0861 0.0000
## [1] -2.0000 0.0000 111.2873 0.0000
## [1] -2.0000 0.1000 105.1005 0.0000
## [1] -2.00000 0.20000 99.57578 0.00000
## [1] -2.00000 0.30000 94.74277 0.00000
## [1] -2.00000 0.40000 90.61234 0.00000
## [1] -2.00000 0.50000 87.17853 0.00000
## [1] -2.00000 0.60000 84.42135 0.00000
## [1] -2.00000 0.70000 82.30982 0.00000
## [1] -2.00000 0.80000 80.80491 0.00000
## [1] -2.0000 0.9000 79.8623 1.0000
## [1] -2.00000 1.00000 79.43468 1.00000
## [1] -2.00000 1.10000 79.47378 1.00000
## [1] -2.00000 1.20000 79.93181 1.00000
## [1] -2.00000 1.30000 80.76263 1.00000
## [1] -2.00000 1.40000 81.92249 1.00000
## [1] -2.00000 1.50000 83.37051 1.00000
## [1] -2.000 1.600 85.069 1.000
## [1] -2.00000 1.70000 86.98346 1.00000
## [1] -2.00000 1.80000 89.08263 1.00000
## [1] -2.00000 1.90000 91.33831 1.00000
## [1] -2.00000 2.00000 93.72523 1.00000
## [1] -2.00000 2.10000 96.22082 1.00000
## [1] -2.00000 2.20000 98.80498 1.00000
## [1] -2.0000 2.3000 101.4599 1.0000
## [1] -2.0000 2.4000 104.1698 1.0000
## [1] -2.0000 2.5000 106.9208 1.0000
## [1] -2.0000 2.6000 109.7005 1.0000
## [1] -2.0000 2.7000 112.4982 1.0000
## [1] -2.0000 2.8000 115.3044 1.0000
## [1] -2.0000 2.9000 118.1107 1.0000
## [1] -2.0000 3.0000 120.9099 1.0000
## [1] -1.900 -1.900 237.743 0.000
## [1] -1.9000 -1.8000 233.9958 0.0000
## [1] -1.9000 -1.7000 229.8648 0.0000
## [1] -1.9000 -1.6000 225.3218 0.0000
## [1] -1.900 -1.500 220.342 0.000
## [1] -1.9000 -1.4000 214.9062 0.0000
## [1] -1.9000 -1.3000 209.0034 0.0000
## [1] -1.9000 -1.2000 202.6336 0.0000
## [1] -1.9000 -1.1000 195.8107 0.0000
## [1] -1.9000 -1.0000 188.5646 0.0000
## [1] -1.9000 -0.9000 180.9435 0.0000
## [1] -1.900 -0.800 173.014 0.000
## [1] -1.9000 -0.7000 164.8609 0.0000
## [1] -1.9000 -0.6000 156.5841 0.0000
## [1] -1.9000 -0.5000 148.2951 0.0000
## [1] -1.9000 -0.4000 140.1118 0.0000
## [1] -1.9000 -0.3000 132.1527 0.0000
## [1] -1.9000 -0.2000 124.5309 0.0000
## [1] -1.9000 -0.1000 117.3487 0.0000
## [1] -1.9000 0.0000 110.6933 0.0000
## [1] -1.9000 0.1000 104.6338 0.0000
## [1] -1.90000 0.20000 99.21972 0.00000
## [1] -1.90000 0.30000 94.48081 0.00000
## [1] -1.90000 0.40000 90.42835 0.00000
## [1] -1.90000 0.50000 87.05721 0.00000
## [1] -1.90000 0.60000 84.34846 0.00000
## [1] -1.90000 0.70000 82.27234 0.00000
## [1] -1.90000 0.80000 80.79112 0.00000
## [1] -1.90000 0.90000 79.86177 1.00000
## [1] -1.90000 1.00000 79.43832 1.00000
## [1] -1.90000 1.10000 79.47368 1.00000
## [1] -1.90000 1.20000 79.92122 1.00000
## [1] -1.90000 1.30000 80.73583 1.00000
## [1] -1.90000 1.40000 81.87468 1.00000
## [1] -1.90000 1.50000 83.29772 1.00000
## [1] -1.90000 1.60000 84.96795 1.00000
## [1] -1.90000 1.70000 86.85153 1.00000
## [1] -1.90000 1.80000 88.91772 1.00000
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## [1] 1.90000 2.10000 77.75533 0.00000
## [1] 1.9000 2.2000 77.7916 0.0000
## [1] 1.90000 2.30000 77.85637 0.00000
## [1] 1.90000 2.40000 77.94808 0.00000
## [1] 1.90000 2.50000 78.06504 0.00000
## [1] 1.90000 2.60000 78.20551 0.00000
## [1] 1.90000 2.70000 78.36768 0.00000
## [1] 1.90000 2.80000 78.54972 0.00000
## [1] 1.90000 2.90000 78.74984 0.00000
## [1] 1.90000 3.00000 78.96625 0.00000
## [1] 2.00000 2.00000 77.75586 0.00000
## [1] 2.0000 2.1000 77.7985 0.0000
## [1] 2.00000 2.20000 77.87518 0.00000
## [1] 2.00000 2.30000 77.98398 0.00000
## [1] 2.00000 2.40000 78.12283 0.00000
## [1] 2.00000 2.50000 78.28959 0.00000
## [1] 2.00000 2.60000 78.48206 0.00000
## [1] 2.00000 2.70000 78.69801 0.00000
## [1] 2.00000 2.80000 78.93526 0.00000
## [1] 2.00000 2.90000 79.19165 0.00000
## [1] 2.00000 3.00000 79.46512 0.00000
## [1] 2.10000 2.10000 77.88204 0.00000
## [1] 2.10000 2.20000 78.00363 0.00000
## [1] 2.10000 2.30000 78.16081 0.00000
## [1] 2.10000 2.40000 78.35098 0.00000
## [1] 2.10000 2.50000 78.57152 0.00000
## [1] 2.10000 2.60000 78.81975 0.00000
## [1] 2.10000 2.70000 79.09306 0.00000
## [1] 2.10000 2.80000 79.38886 0.00000
## [1] 2.10000 2.90000 79.70468 0.00000
## [1] 2.10000 3.00000 80.03812 0.00000
## [1] 2.20000 2.20000 78.17401 0.00000
## [1] 2.20000 2.30000 78.38328 0.00000
## [1] 2.2000 2.4000 78.6283 0.0000
## [1] 2.20000 2.50000 78.90591 0.00000
## [1] 2.20000 2.60000 79.21302 0.00000
## [1] 2.20000 2.70000 79.54656 0.00000
## [1] 2.20000 2.80000 79.90359 0.00000
## [1] 2.2000 2.9000 80.2813 0.0000
## [1] 2.200 3.000 80.677 0.000
## [1] 2.30000 2.30000 78.64774 0.00000
## [1] 2.30000 2.40000 78.95049 0.00000
## [1] 2.30000 2.50000 79.28788 0.00000
## [1] 2.30000 2.60000 79.65633 0.00000
## [1] 2.30000 2.70000 80.05238 0.00000
## [1] 2.30000 2.80000 80.47271 0.00000
## [1] 2.30000 2.90000 80.91418 0.00000
## [1] 2.30000 3.00000 81.37383 0.00000
## [1] 2.40000 2.40000 79.31332 0.00000
## [1] 2.4000 2.5000 79.7126 0.0000
## [1] 2.40000 2.60000 80.14432 0.00000
## [1] 2.40000 2.70000 80.60459 0.00000
## [1] 2.40000 2.80000 81.08976 0.00000
## [1] 2.40000 2.90000 81.59635 0.00000
## [1] 2.40000 3.00000 82.12115 0.00000
## [1] 2.50000 2.50000 80.17541 0.00000
## [1] 2.50000 2.60000 80.67181 0.00000
## [1] 2.50000 2.70000 81.19756 0.00000
## [1] 2.50000 2.80000 81.74862 0.00000
## [1] 2.50000 2.90000 82.32124 0.00000
## [1] 2.50000 3.00000 82.91193 0.00000
## [1] 2.6000 2.6000 81.2339 0.0000
## [1] 2.60000 2.70000 81.82592 0.00000
## [1] 2.60000 2.80000 82.44353 0.00000
## [1] 2.60000 2.90000 83.08268 0.00000
## [1] 2.60000 3.00000 83.73965 0.00000
## [1] 2.70000 2.70000 82.48466 0.00000
## [1] 2.70000 2.80000 83.16913 0.00000
## [1] 2.70000 2.90000 83.87499 0.00000
## [1] 2.70000 3.00000 84.59829 0.00000
## [1] 2.80000 2.80000 83.92044 0.00000
## [1] 2.8000 2.9000 84.6929 0.0000
## [1] 2.80000 3.00000 85.48231 0.00000
## [1] 2.90000 2.90000 85.53159 0.00000
## [1] 2.90000 3.00000 86.38668 0.00000
## [1] 3.0000 3.0000 87.3068 0.0000## $power
## [1] 1.6 2.1
##
## $deviance
## [1] 77.88717
##
## $mistake
## NULLPlot using the obtained power 1.6 and 2.1.
Age<-hav$Age
pos<-hav$Pos
Tot<-hav$Tot
model.fp2.m<-glm(cbind(pos,Tot-pos) ~ I(Age^1.6) + I(Age^2.1), family=binomial(link="logit"))
par(las=1,cex.axis=1.1,cex.lab=1.1,lwd=3,mgp=c(2, 0.5, 0),mar=c(4.1,4.1,4.1,3))
plot(Age,pos/Tot,cex=0.1*Tot,xlab="age",ylab="seroprevalence",xlim=c(0,86),ylim=c(-0.2,1),yaxt="n")
lines(Age,model.fp2.m$fitted.values,lwd=2,lty=3,col='orange')
foi.num<-function(x,p) # p fitted value
{
grid<-sort(unique(x))
pgrid<-(p[order(x)])[duplicated(sort(x))==F]
dp<-diff(pgrid)/diff(grid)
foi<-approx((grid[-1]+grid[-length(grid)])/2,dp,grid[c(-1,-length(grid))])$y/(1-pgrid[c(-1,-length(grid))]) #y of approx function
return(list(grid=grid[c(-1,-length(grid))],foi=foi))
}
model.fp2.m.foi<-foi.num(Age,model.fp2.m$fitted.values)
lines(model.fp2.m.foi$grid,4*model.fp2.m.foi$foi,lwd=2,lty=3,col='orange')
abline(0,0,lwd=2)
axis(side=4,at=c(0.0,0.2,0.4),labels=c(0.00,0.05,0.10))
axis(side=2,at=seq(0,1,0.2),labels=seq(0,1,0.2))
mtext(side=4,"force of infection", las=3,line=2)
Now we see the foi is above the zero.
The key feature of the fractional polynomial regression is we use the command
diff(predict(glm.try))to check if the predicted values are increasing, in fact we also can use the command
diff(fitted(glm.try))to check if the prevalence was increasing with age, we should get the same results.
Note in R, predict function returns the fitted values before the inverse of the link function is applied, and fitted function returns values after it is applied. such as in a logistic regression predicted function will return \(\eta\) while fitted function will return the \(p\) i.e the probability of an event.
References
1.Hens, Niel, et al. Modeling infectious disease parameters based on serological and social contact data: a modern statistical perspective. Springer Science & Business Media, 2012.
2.Shkedy, Ziv, et al. “Modelling forces of infection by using monotone local polynomials.” Journal of the Royal Statistical Society Series C: Applied Statistics 52.4 (2003): 469-485.