# Biostatistics III in R

#### Code by Johan Zetterqvist, Andreas Karlsson and Mark Clements

# Exercise 6. Diet data: tabulating incidence rates and modelling with Poisson regression

You may have to install the required packages the first time you use them. You can install a package by `install.packages("package_of_interest")` for each package you require.

    library(biostat3)
    library(dplyr)    # for data manipulation

Load the diet data using time-on-study as the timescale.

    head(diet)

    ##    id chd         y hieng  energy       job month   height   weight
    ## 1 127   0 16.791239   low 2023.25 conductor     2 173.9900 61.46280
    ## 2 200   0 19.958933   low 2448.68      bank    12 177.8000 73.48320
    ## 3 198   0 19.958933   low 2281.38      bank    12       NA       NA
    ## 4 222   0 15.394935   low 2467.95      bank     2 158.7500 58.24224
    ## 5 305   1  1.494866   low 2362.93      bank     1       NA       NA
    ## 6 173   0 15.958932   low 2380.11 conductor    12 164.4904 79.01712
    ##          doe        dox        dob      yoe      yox      yob         y1k
    ## 1 1960-02-16 1976-12-01 1910-09-27 1960.126 1976.917 1910.737 0.016791239
    ## 2 1956-12-16 1976-12-01 1909-06-18 1956.958 1976.917 1909.460 0.019958933
    ## 3 1956-12-16 1976-12-01 1910-06-30 1956.958 1976.917 1910.493 0.019958933
    ## 4 1957-02-16 1972-07-10 1902-07-11 1957.126 1972.523 1902.523 0.015394935
    ## 5 1960-01-16 1961-07-15 1913-06-30 1960.041 1961.534 1913.493 0.001494866
    ## 6 1960-12-16 1976-12-01 1915-06-28 1960.958 1976.917 1915.487 0.015958932
    ##   eng3 X1 X2 X3 eng3low eng3medium eng3high
    ## 1  low  1  0  0       1          0        0
    ## 2  low  1  0  0       1          0        0
    ## 3  low  1  0  0       1          0        0
    ## 4  low  1  0  0       1          0        0
    ## 5  low  1  0  0       1          0        0
    ## 6  low  1  0  0       1          0        0

    summary(diet)

    ##        id           chd               y            hieng
    ##  Min.   :  1   Min.   :0.0000   Min.   : 0.2875   low :155
    ##  1st Qu.: 85   1st Qu.:0.0000   1st Qu.:10.7762   high:182
    ##  Median :169   Median :0.0000   Median :15.4606
    ##  Mean   :169   Mean   :0.1365   Mean   :13.6607
    ##  3rd Qu.:253   3rd Qu.:0.0000   3rd Qu.:17.0431
    ##  Max.   :337   Max.   :1.0000   Max.   :20.0411
    ##
    ##      energy            job          month            height
    ##  Min.   :1748   driver   :102   Min.   : 1.000   Min.   :152.4
    ##  1st Qu.:2537   conductor: 84   1st Qu.: 3.000   1st Qu.:168.9
    ##  Median :2803   bank     :151   Median : 6.000   Median :173.0
    ##  Mean   :2829                   Mean   : 6.231   Mean   :173.4
    ##  3rd Qu.:3110                   3rd Qu.:10.000   3rd Qu.:177.8
    ##  Max.   :4396                   Max.   :12.000   Max.   :190.5
    ##                                                  NA's   :5
    ##      weight            doe                  dox
    ##  Min.   : 46.72   Min.   :1956-11-16   Min.   :1958-08-29
    ##  1st Qu.: 64.64   1st Qu.:1959-01-16   1st Qu.:1972-09-29
    ##  Median : 72.80   Median :1960-02-16   Median :1976-12-01
    ##  Mean   : 72.54   Mean   :1960-06-22   Mean   :1974-02-19
    ##  3rd Qu.: 79.83   3rd Qu.:1961-06-16   3rd Qu.:1976-12-01
    ##  Max.   :106.14   Max.   :1966-09-16   Max.   :1976-12-01
    ##  NA's   :4
    ##       dob                  yoe            yox            yob
    ##  Min.   :1892-01-10   Min.   :1957   Min.   :1959   Min.   :1892
    ##  1st Qu.:1906-01-18   1st Qu.:1959   1st Qu.:1973   1st Qu.:1906
    ##  Median :1911-02-25   Median :1960   Median :1977   Median :1911
    ##  Mean   :1911-01-04   Mean   :1960   Mean   :1974   Mean   :1911
    ##  3rd Qu.:1915-01-30   3rd Qu.:1961   3rd Qu.:1977   3rd Qu.:1915
    ##  Max.   :1930-09-19   Max.   :1967   Max.   :1977   Max.   :1931
    ##
    ##       y1k                eng3           X1               X2
    ##  Min.   :0.0002875   low   : 75   Min.   :0.0000   Min.   :0.0000
    ##  1st Qu.:0.0107762   medium:150   1st Qu.:0.0000   1st Qu.:0.0000
    ##  Median :0.0154606   high  :112   Median :0.0000   Median :0.0000
    ##  Mean   :0.0136607                Mean   :0.2226   Mean   :0.4451
    ##  3rd Qu.:0.0170431                3rd Qu.:0.0000   3rd Qu.:1.0000
    ##  Max.   :0.0200411                Max.   :1.0000   Max.   :1.0000
    ##
    ##        X3            eng3low         eng3medium        eng3high
    ##  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
    ##  1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000
    ##  Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000
    ##  Mean   :0.3323   Mean   :0.2226   Mean   :0.4451   Mean   :0.3323
    ##  3rd Qu.:1.0000   3rd Qu.:0.0000   3rd Qu.:1.0000   3rd Qu.:1.0000
    ##  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000
    ##

## (a)

    diet <- biostat3::diet
    diet$y1k <- diet$y/1000

    diet.ir6a <- survRate(Surv(y/1000,chd) ~ hieng, data=diet)
    ## or
    diet %>%
        group_by(hieng) %>%
        summarise(Event = sum(chd), Time = sum(y1k), Rate = Event/Time,      # group sums
                  CI_low = poisson.test(Event,Time)$conf.int[1],
                  CI_high = poisson.test(Event,Time)$conf.int[2])

    ## # A tibble: 2 x 6
    ##    hieng Event     Time      Rate   CI_low  CI_high
    ##   <fctr> <int>    <dbl>     <dbl>    <dbl>    <dbl>
    ## 1    low    28 2.059431 13.595992 9.034438 19.64999
    ## 2   high    18 2.544238  7.074809 4.192980 11.18125

    ## IRR
    with(diet.ir6a, poisson.test(event,tstop))

    ##
    ##  Comparison of Poisson rates
    ##
    ## data:  event time base: tstop
    ## count1 = 28, expected count1 = 20.578, p-value = 0.03681
    ## alternative hypothesis: true rate ratio is not equal to 1
    ## 95 percent confidence interval:
    ##  1.026173 3.688904
    ## sample estimates:
    ## rate ratio
    ##   1.921747

    with(diet.ir6a, poisson.test(rev(event),rev(tstop)))

    ##
    ##  Comparison of Poisson rates
    ##
    ## data:  rev(event) time base: rev(tstop)
    ## count1 = 18, expected count1 = 25.422, p-value = 0.03681
    ## alternative hypothesis: true rate ratio is not equal to 1
    ## 95 percent confidence interval:
    ##  0.2710832 0.9744943
    ## sample estimates:
    ## rate ratio
    ##  0.5203599

We see that individuals with a high energy intake have a lower CHD incidence rate. The estimated crude incidence rate ratio is 0.52 (95% CI: 0.27, 0.97).

## (b)

    poisson6b <- glm( chd ~ hieng + offset( log( y1k ) ), family=poisson, data=diet)
    summary(poisson6b)

    ##
    ## Call:
    ## glm(formula = chd ~ hieng + offset(log(y1k)), family = poisson,
    ##     data = diet)
    ##
    ## Deviance Residuals:
    ##     Min       1Q   Median       3Q      Max
    ## -0.7382  -0.6337  -0.4899  -0.3891   3.0161
    ##
    ## Coefficients:
    ##             Estimate Std. Error z value Pr(>|z|)
    ## (Intercept)   2.6098     0.1890  13.811   <2e-16 ***
    ## hienghigh    -0.6532     0.3021  -2.162   0.0306 *
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## (Dispersion parameter for poisson family taken to be 1)
    ##
    ##     Null deviance: 262.82  on 336  degrees of freedom
    ## Residual deviance: 258.00  on 335  degrees of freedom
    ## AIC: 354
    ##
    ## Number of Fisher Scoring iterations: 6

    eform(poisson6b)

    ##              exp(beta)     2.5 %     97.5 %
    ## (Intercept) 13.5959916 9.3877130 19.6907371
    ## hienghigh    0.5203599 0.2878432  0.9407012

    eform(poisson6b, method="Profile")

    ## Waiting for profiling to be done...

    ##              exp(beta)     2.5 %     97.5 %
    ## (Intercept) 13.5959916 9.1614552 19.2715805
    ## hienghigh    0.5203599 0.2829171  0.9328392

The point estimate for the IRR calculated by the Poisson regression is the same as the IRR calculated in 6(a). A theoretical observation: if we consider the data as being cross-classified solely by `hieng` then the Poisson regression model with one parameter is a saturated model so the IRR estimated from the model will be identical to the ‘observed’ IRR. That is, the model is a perfect fit.

## (c)

    hist6c <- hist(diet$energy, breaks=25, probability=TRUE, xlab="Energy (units)")
    curve(dnorm(x, mean=mean(diet$energy), sd=sd(diet$energy)), col = "red", add=TRUE)

    quantile(diet$energy, probs=c(0.01,0.05,0.1,0.25,0.5,0.75,0.90,0.95,0.99))

    ##       1%       5%      10%      25%      50%      75%      90%      95%
    ## 1887.268 2177.276 2314.114 2536.690 2802.980 3109.660 3365.644 3588.178
    ##      99%
    ## 4046.820

    # For kurtosis and skewness, see package e1071

The histogram gives us an idea of the distribution of energy intake. We can also tabulate moments and percentiles of the distribution.

## (d)

    diet$eng3 <- cut(diet$energy, breaks=c(1500,2500,3000,4500),labels=c("low","medium","high"),
                     right = FALSE)
    cbind(Freq=table(diet$eng3),
          Prop=table(diet$eng3)/nrow(diet))

    ##        Freq      Prop
    ## low      75 0.2225519
    ## medium  150 0.4451039
    ## high    112 0.3323442

## (e)

    ## rates and IRRs
    diet.ir6e <- survRate(Surv(y/1000,chd) ~ eng3, data=diet)
    print(diet.ir6e)

    ##               eng3     tstop event      rate    lower     upper
    ## eng3=low       low 0.9466338    16 16.901995 9.660951 27.447781
    ## eng3=medium medium 2.0172621    22 10.905871 6.834651 16.511619
    ## eng3=high     high 1.6397728     8  4.878725 2.106287  9.613033

    # calculate IRR and confidence intervals
    with(diet.ir6e, rate[eng3=="medium"] / rate[eng3=="low"])

    ## [1] 0.6452416

    with(diet.ir6e[c(2,1),], { # compare second row with first row
      poisson.test(event, tstop)
    })

    ##
    ##  Comparison of Poisson rates
    ##
    ## data:  event time base: tstop
    ## count1 = 22, expected count1 = 25.863, p-value = 0.2221
    ## alternative hypothesis: true rate ratio is not equal to 1
    ## 95 percent confidence interval:
    ##  0.3237007 1.3143509
    ## sample estimates:
    ## rate ratio
    ##  0.6452416

    with(diet.ir6e, rate[eng3=="high"] / rate[eng3=="low"])

    ## [1] 0.2886479

    with(diet.ir6e[c(3,1),], { # compare third row with first row
      poisson.test(event, tstop)
    })

    ##
    ##  Comparison of Poisson rates
    ##
    ## data:  event time base: tstop
    ## count1 = 8, expected count1 = 15.216, p-value = 0.004579
    ## alternative hypothesis: true rate ratio is not equal to 1
    ## 95 percent confidence interval:
    ##  0.1069490 0.7148284
    ## sample estimates:
    ## rate ratio
    ##  0.2886479

We see that the CHD incidence rate decreases as the level of total energy intake increases.

## (f)

    diet <- mutate(diet,
                   X1 = as.numeric(eng3 == "low"),
                   X2 = as.numeric(eng3 == "medium"),
                   X3 = as.numeric(eng3 == "high"))
    # or
    diet <- biostat3::addIndicators(diet, ~eng3+0) %>%
        mutate(X1 = eng3low, X2 = eng3medium, X3 = eng3high)
    colSums(diet[c("X1","X2","X3")])

    ##  X1  X2  X3
    ##  75 150 112

## (g)

    filter(diet, eng3=="low")    %>% select(c(energy,eng3,X1,X2,X3)) %>% head

    ##    energy eng3 X1 X2 X3
    ## 1 2023.25  low  1  0  0
    ## 2 2448.68  low  1  0  0
    ## 3 2281.38  low  1  0  0
    ## 4 2467.95  low  1  0  0
    ## 5 2362.93  low  1  0  0
    ## 6 2380.11  low  1  0  0

    filter(diet, eng3=="medium") %>% select(c(energy,eng3,X1,X2,X3)) %>% head

    ##    energy   eng3 X1 X2 X3
    ## 1 2664.64 medium  0  1  0
    ## 2 2533.33 medium  0  1  0
    ## 3 2854.08 medium  0  1  0
    ## 4 2673.77 medium  0  1  0
    ## 5 2766.88 medium  0  1  0
    ## 6 2586.69 medium  0  1  0

    filter(diet, eng3=="high")   %>% select(c(energy,eng3,X1,X2,X3)) %>% head

    ##    energy eng3 X1 X2 X3
    ## 1 3067.36 high  0  0  1
    ## 2 3298.95 high  0  0  1
    ## 3 3147.60 high  0  0  1
    ## 4 3180.47 high  0  0  1
    ## 5 3045.81 high  0  0  1
    ## 6 3060.03 high  0  0  1

## (h)

    poisson6h <- glm( chd ~ X2 + X3 + offset( log( y1k ) ), family=poisson, data=diet )
    summary(poisson6h)

    ##
    ## Call:
    ## glm(formula = chd ~ X2 + X3 + offset(log(y1k)), family = poisson,
    ##     data = diet)
    ##
    ## Deviance Residuals:
    ##     Min       1Q   Median       3Q      Max
    ## -0.8231  -0.6052  -0.4532  -0.3650   2.9434
    ##
    ## Coefficients:
    ##             Estimate Std. Error z value Pr(>|z|)
    ## (Intercept)   2.8274     0.2500  11.312  < 2e-16 ***
    ## X2           -0.4381     0.3285  -1.334  0.18233
    ## X3           -1.2425     0.4330  -2.870  0.00411 **
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## (Dispersion parameter for poisson family taken to be 1)
    ##
    ##     Null deviance: 262.82  on 336  degrees of freedom
    ## Residual deviance: 253.62  on 334  degrees of freedom
    ## AIC: 351.62
    ##
    ## Number of Fisher Scoring iterations: 6

    eform(poisson6h)

    ##              exp(beta)      2.5 %     97.5 %
    ## (Intercept) 16.9019960 10.3556032 27.5867534
    ## X2           0.6452416  0.3389050  1.2284761
    ## X3           0.2886478  0.1235407  0.6744143

Level 1 of the categorized total energy is the reference category. The estimated rate ratio comparing level 2 to level 1 is 0.6452 and the estimated rate ratio comparing level 3 to level 1 is 0.2886.

## (i)

    poisson6i <- glm( chd ~ X1 + X3 + offset( log( y1k ) ), family=poisson, data=diet )
    # or
    poisson6i <- glm( chd ~ I(eng3=="low") + I(eng3=="high") + offset( log( y1k ) ), family=poisson, data=diet )

    summary( poisson6i )

    ##
    ## Call:
    ## glm(formula = chd ~ I(eng3 == "low") + I(eng3 == "high") + offset(log(y1k)),
    ##     family = poisson, data = diet)
    ##
    ## Deviance Residuals:
    ##     Min       1Q   Median       3Q      Max
    ## -0.8231  -0.6052  -0.4532  -0.3650   2.9434
    ##
    ## Coefficients:
    ##                       Estimate Std. Error z value Pr(>|z|)
    ## (Intercept)             2.3893     0.2132  11.207   <2e-16 ***
    ## I(eng3 == "low")TRUE    0.4381     0.3285   1.334   0.1823
    ## I(eng3 == "high")TRUE  -0.8044     0.4129  -1.948   0.0514 .
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## (Dispersion parameter for poisson family taken to be 1)
    ##
    ##     Null deviance: 262.82  on 336  degrees of freedom
    ## Residual deviance: 253.62  on 334  degrees of freedom
    ## AIC: 351.62
    ##
    ## Number of Fisher Scoring iterations: 6

    eform( poisson6i )

    ##                        exp(beta)     2.5 %    97.5 %
    ## (Intercept)           10.9058706 7.1810131 16.562846
    ## I(eng3 == "low")TRUE   1.5498071 0.8140167  2.950679
    ## I(eng3 == "high")TRUE  0.4473485 0.1991681  1.004783

Now use level 2 as the reference (by omitting X2 but including X1 and X3). The estimated rate ratio comparing level 1 to level 2 is 1.5498 and the estimated rate ratio comparing level 3 to level 2 is 0.4473.

## (j)

    poisson6j <- glm( chd ~ eng3 + offset( log( y1k ) ), family=poisson, data=diet )
    summary( poisson6j )

    ##
    ## Call:
    ## glm(formula = chd ~ eng3 + offset(log(y1k)), family = poisson,
    ##     data = diet)
    ##
    ## Deviance Residuals:
    ##     Min       1Q   Median       3Q      Max
    ## -0.8231  -0.6052  -0.4532  -0.3650   2.9434
    ##
    ## Coefficients:
    ##             Estimate Std. Error z value Pr(>|z|)
    ## (Intercept)   2.8274     0.2500  11.312  < 2e-16 ***
    ## eng3medium   -0.4381     0.3285  -1.334  0.18233
    ## eng3high     -1.2425     0.4330  -2.870  0.00411 **
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## (Dispersion parameter for poisson family taken to be 1)
    ##
    ##     Null deviance: 262.82  on 336  degrees of freedom
    ## Residual deviance: 253.62  on 334  degrees of freedom
    ## AIC: 351.62
    ##
    ## Number of Fisher Scoring iterations: 6

    eform( poisson6j )

    ##              exp(beta)      2.5 %     97.5 %
    ## (Intercept) 16.9019960 10.3556032 27.5867534
    ## eng3medium   0.6452416  0.3389050  1.2284761
    ## eng3high     0.2886478  0.1235407  0.6744143

The estimates are identical (as we would hope) when we have R create indicator variables for us.

## (k)

Somehow (there are many different alternatives) you’ll need to calculate the total number of events and the total person-time at risk and then calculate the incidence rate as events/person-time. For example,

    summarise(diet, rate = sum(chd) / sum(y))

    ##          rate
    ## 1 0.009992031

The estimated incidence rate is 0.00999 events per person-year.