The aim of this vignette is to illustrate the use/functionality of the `glm_coef`

function. `glm_coef`

can be used to display model coefficients with confidence intervals and p-values. The advantages and limitations of `glm_coef`

are:

- Recognises the main models used in epidemiology/public health.
- Automatically back-transforms estimates and confidence intervals, when the model requires it.
- Can use robust standard errors for the calculation of confidence intervals.
- Standard errors are used by default.
- The use of standard errors is restricted by the following classes of objects (models):
`gee`

,`glm`

and`survreg`

.

- Can display nice labels for the names of the parameters.
- Returns a data frame that can be modified and/or exported as tables for publications (with further editing).

We start by loading relevant packages and setting the theme for the plots (as suggested in the Template of this package):

```
rm(list = ls())
library(car)
library(broom)
library(kableExtra)
library(tidyverse)
library(ggfortify)
library(mosaic)
library(huxtable)
library(jtools)
library(latex2exp)
library(pubh)
library(sjlabelled)
library(sjPlot)
library(sjmisc)
theme_set(sjPlot::theme_sjplot2(base_size = 10))
theme_update(legend.position = "top")
# options('huxtable.knit_print_df' = FALSE)
options('huxtable.knit_print_df_theme' = theme_article)
options('huxtable.autoformat_number_format' = list(numeric = "%5.2f"))
knitr::opts_chunk$set(collapse = TRUE, comment = NA)
```

For continuous outcomes there is no need of exponentiating the results unless the outcome was fitted in the log-scale. In our first example we want to estimate the effect of smoking and race on the birth weight of babies.

We can generate factors and assign labels in the same `pipe`

stream:

```
data(birthwt, package = "MASS")
birthwt <- birthwt %>%
mutate(
smoke = factor(smoke, labels = c("Non-smoker", "Smoker")),
race = factor(race, labels = c("White", "African American", "Other"))
) %>%
var_labels(
bwt = 'Birth weight (g)',
smoke = 'Smoking status',
race = 'Race'
)
```

Is good to start with some basic descriptive statistics, so we can compare the birth weight between groups.

```
birthwt %>%
group_by(race, smoke) %>%
summarise(
n = n(),
Mean = mean(bwt, na.rm = TRUE),
SD = sd(bwt, na.rm = TRUE),
Median = median(bwt, na.rm = TRUE),
CV = rel_dis(bwt)
)
# A tibble: 6 x 7
# Groups: race [3]
race smoke n Mean SD Median CV
<fct> <fct> <int> <dbl> <dbl> <dbl> <dbl>
1 White Non-smoker 44 3429. 710. 3593 0.207
2 White Smoker 52 2827. 626. 2776. 0.222
3 African American Non-smoker 16 2854. 621. 2920 0.218
4 African American Smoker 10 2504 637. 2381 0.254
5 Other Non-smoker 55 2816. 709. 2807 0.252
6 Other Smoker 12 2757. 810. 3146. 0.294
```

From the previous table, the group with the lower birth weight was from babies born from African Americans who were smokers. The highest birth weight was from babies born from White non-smokers.

Another way to compare the means between the groups is with `gen_bst_df`

which estimates means with corresponding bootstrapped CIs.

Birth weight (g) | LowerCI | UpperCI | Race | Smoking status |

3428.75 | 3197.46 | 3640.79 | White | Non-smoker |

2826.85 | 2659.80 | 3002.00 | White | Smoker |

2854.50 | 2529.15 | 3145.42 | African American | Non-smoker |

2504.00 | 2102.15 | 2891.36 | African American | Smoker |

2815.78 | 2628.10 | 2990.50 | Other | Non-smoker |

2757.17 | 2302.05 | 3169.93 | Other | Smoker |

Another approach to tabular analysis is graphical analysis. For this comparison, box-plots would be the way to go, but in health sciences it is more common to see bar charts with error bars.

```
birthwt %>%
bar_error(bwt ~ race, fill = ~ smoke) %>%
axis_labs() %>%
gf_labs(fill = "Smoking status:")
No summary function supplied, defaulting to `mean_se()`
```

We fit a linear model.

Note:Model diagnostics are not be discussed in this vignette.

Traditional output from the model:

term | sumsq | df | statistic | p.value |

smoke | 7322574.73 | 1.00 | 15.46 | 0.00 |

race | 8712354.03 | 2.00 | 9.20 | 0.00 |

Residuals | 87631355.83 | 185.00 |

term | estimate | std.error | statistic | p.value |

(Intercept) | 3334.95 | 91.78 | 36.34 | 0.00 |

smokeSmoker | -428.73 | 109.04 | -3.93 | 0.00 |

raceAfrican American | -450.36 | 153.12 | -2.94 | 0.00 |

raceOther | -452.88 | 116.48 | -3.89 | 0.00 |

Table of coefficients:

Parameter | Coefficient | Pr(>|t|) |

Constant | 3334.95 (3036.11, 3633.78) | < 0.001 |

Smoking status: Smoker | -428.73 (-667.02, -190.44) | < 0.001 |

Race: African American | -450.36 (-750.31, -150.4) | 0.003 |

Race: Other | -452.88 (-775.17, -130.58) | 0.006 |

Note:Compare results using naive standard errors.

Parameter | Coefficient | Pr(>|t|) |

Constant | 3334.95 (3153.89, 3516.01) | < 0.001 |

Smoking status: Smoker | -428.73 (-643.86, -213.6) | < 0.001 |

Race: African American | -450.36 (-752.45, -148.27) | 0.004 |

Race: Other | -452.88 (-682.67, -223.08) | < 0.001 |

The function `glance`

from the `broom`

package allow us to have a quick look at statistics related with the model.

r.squared | adj.r.squared | sigma | statistic | p.value | df | logLik | AIC | BIC | deviance | df.residual |

0.12 | 0.11 | 688.25 | 8.68 | 0.00 | 4 | -1501.11 | 3012.22 | 3028.43 | 87631355.83 | 185 |

To construct the effect plot, we can use `plot_model`

from the `sjPlot`

package. The advantage of `plot_model`

is that recognises labelled data and uses that information for annotating the plots.

When the explanatory variables are categorical, another option is `emmip`

from the `emmeans`

package. We can include CIs in `emmip`

but as estimates are *connected*, the resulting plots look more messy, so I recommend `emmip`

to look at the trace.

For logistic regression we are interested in the odds ratios. We will look at the effect of amount of fibre intake on the development of coronary heart disease.

```
data(diet, package = "Epi")
diet <- diet %>%
mutate(
chd = factor(chd, labels = c("No CHD", "CHD"))
) %>%
var_labels(
chd = "Coronary Heart Disease",
fibre = "Fibre intake (10 g/day)"
)
```

We start with descriptive statistics:

Coronary Heart Disease | N | Min. | Max. | Mean | Median | SD | CV | |

Fibre intake (10 g/day) | No CHD | 288.00 | 0.60 | 5.35 | 1.75 | 1.69 | 0.58 | 0.33 |

CHD | 45.00 | 0.76 | 2.43 | 1.49 | 1.51 | 0.40 | 0.27 |

It is standard to plot the dependent variable in the \(y\)-axis, so in this case, we can use horizontal box-plots.

```
diet %>%
gf_boxploth(chd ~ fibre, fill = "indianred3", alpha = 0.7) %>%
axis_labs()
Warning: Removed 4 rows containing non-finite values (stat_boxploth).
```

We fit a linear logistic model:

```
model_binom <- glm(chd ~ fibre, data = diet, family = binomial)
model_binom %>%
glm_coef(labels = model_labels(model_binom))
```

Parameter | Odds ratio | Pr(>|z|) |

Constant | 0.95 (0.3, 3.01) | 0.93 |

Fibre intake (10 g/day) | 0.33 (0.16, 0.67) | 0.00 |

Effect plot:

We will look at a matched case-control study on the effect of oestrogen use and history of gall bladder disease on the development of endometrial cancer.

```
data(bdendo, package = "Epi")
bdendo <- bdendo %>%
mutate(
cancer = factor(d, labels = c('Control', 'Case')),
gall = factor(gall, labels = c("No GBD", "GBD")),
est = factor(est, labels = c("No oestrogen", "Oestrogen"))
) %>%
var_labels(
cancer = 'Endometrial cancer',
gall = 'Gall bladder disease',
est = 'Oestrogen'
)
```

We start with descriptive statistics:

Endometrial cancer | levels | Control | Case | Total |

Total N (%) | 252 (80.0) | 63 (20.0) | 315 | |

Oestrogen | No oestrogen | 125 (49.6) | 7 (11.1) | 132 (41.9) |

Oestrogen | 127 (50.4) | 56 (88.9) | 183 (58.1) | |

Gall bladder disease | No GBD | 228 (90.5) | 46 (73.0) | 274 (87.0) |

GBD | 24 (9.5) | 17 (27.0) | 41 (13.0) |

We fit the conditional logistic model:

```
library(survival)
model_clogit <- clogit(cancer == 'Case' ~ est * gall + strata(set), data = bdendo)
model_clogit %>%
glm_coef(labels = c("Oestrogen/No oestrogen", "GBD/No GBD",
"Oestrogen:GBD Interaction"))
```

Parameter | Odds ratio | Pr(>|z|) |

Oestrogen/No oestrogen | 14.88 (4.49, 49.36) | < 0.001 |

GBD/No GBD | 18.07 (3.2, 102.01) | 0.001 |

Oestrogen:GBD Interaction | 0.13 (0.02, 0.9) | 0.039 |

Creating data frame needed to construct the effect plot:

```
require(ggeffects)
Loading required package: ggeffects
bdendo_pred <- ggemmeans(model_clogit, terms = c('gall', 'est'))
```

Effect plot:

We use data about house satisfaction.

```
library(ordinal)
Attaching package: 'ordinal'
The following object is masked from 'package:dplyr':
slice
data(housing, package = "MASS")
housing <- housing %>%
var_labels(
Sat = "Satisfaction",
Infl = "Perceived influence",
Type = "Type of rental",
Cont = "Contact"
)
```

We fit the ordinal logistic model:

Parameter | Ordinal OR | Pr(>|Z|) |

Perceived influence: Low | 0.61 (0.48, 0.78) | < 0.001 |

Perceived influence: Medium | 2 (1.56, 2.55) | < 0.001 |

Contact: Low | 1.76 (1.44, 2.16) | < 0.001 |

Perceived influence: High | 3.63 (2.83, 4.66) | < 0.001 |

Contact: High | 0.56 (0.45, 0.71) | < 0.001 |

Type of rental: Apartment | 0.69 (0.51, 0.94) | 0.018 |

Type of rental: Atrium | 0.34 (0.25, 0.45) | < 0.001 |

Type of rental: Terrace | 1.43 (1.19, 1.73) | < 0.001 |

Effect plots:

```
plot_model(model_clm, type = "pred", terms = c("Infl", "Cont"),
dot.size = 1, title = "") %>%
gf_theme(axis.text.x = element_text(angle = 45, hjust = 1))
```

```
plot_model(model_clm, type = "pred", terms = c("Infl", "Type"),
dot.size = 1, title = "") %>%
gf_theme(axis.text.x = element_text(angle = 45, hjust = 1))
```

Note:In the previous table parameter estimates and confidence intervals forPerceived influenceandAccommodationwere not adjusted for multiple comparisons.

For Poisson regression we are interested in incidence rate ratios. We will look at the effect of sex, ethnicity and age group on number of absent days from school in a year.

```
data(quine, package = "MASS")
levels(quine$Eth) <- c("Aboriginal", "White")
levels(quine$Sex) <- c("Female", "Male")
```

```
quine <- quine %>%
var_labels(
Days = "Number of absent days",
Eth = "Ethnicity",
Age = "Age group"
)
```

Descriptive statistics:

```
quine %>%
group_by(Eth, Sex, Age) %>%
summarise(
n = n(),
Mean = mean(Days, na.rm = TRUE),
SD = sd(Days, na.rm = TRUE),
Median = median(Days, na.rm = TRUE),
CV = rel_dis(Days)
)
# A tibble: 16 x 8
# Groups: Eth, Sex [4]
Eth Sex Age n Mean SD Median CV
<fct> <fct> <fct> <int> <dbl> <dbl> <dbl> <dbl>
1 Aboriginal Female F0 5 17.6 17.4 11 0.987
2 Aboriginal Female F1 15 18.9 16.3 13 0.865
3 Aboriginal Female F2 9 32.6 27.3 20 0.839
4 Aboriginal Female F3 9 14.6 14.9 10 1.02
5 Aboriginal Male F0 8 11.5 7.23 12 0.629
6 Aboriginal Male F1 5 9.6 4.51 7 0.469
7 Aboriginal Male F2 11 30.9 17.8 32 0.576
8 Aboriginal Male F3 7 27.1 10.4 28 0.382
9 White Female F0 5 19.8 9.68 20 0.489
10 White Female F1 17 7.76 6.48 6 0.834
11 White Female F2 10 5.7 4.97 4 0.872
12 White Female F3 10 13.5 11.5 12 0.851
13 White Male F0 9 13.6 20.9 7 1.54
14 White Male F1 9 5.56 5.39 5 0.970
15 White Male F2 10 15.2 12.9 12 0.848
16 White Male F3 7 27.3 22.9 27 0.840
```

We start by fitting a standard Poisson linear regression model:

```
model_pois <- glm(Days ~ Eth + Sex + Age, family = poisson, data = quine)
model_pois %>%
glm_coef(labels = model_labels(model_pois))
```

Parameter | Rate ratio | Pr(>|z|) |

Constant | 17.66 (10.66, 29.24) | < 0.001 |

Ethnicity: White | 0.59 (0.39, 0.88) | 0.01 |

Sex: Male | 1.11 (0.77, 1.6) | 0.57 |

Age group: F1 | 0.8 (0.42, 1.5) | 0.482 |

Age group: F2 | 1.42 (0.85, 2.36) | 0.181 |

Age group: F3 | 1.35 (0.77, 2.34) | 0.292 |

null.deviance | df.null | logLik | AIC | BIC | deviance | df.residual |

2073.53 | 145 | -1165.49 | 2342.98 | 2360.88 | 1742.50 | 140 |

The assumption is that the mean is equal than the variance. If that is the case, deviance should be close to the degrees of freedom of the residuals (look at the above output from `glance`

). In other words, the following calculation should be close to 1:

Thus, we have over-dispersion. One option is to use a negative binomial distribution.

```
library(MASS)
Attaching package: 'MASS'
The following objects are masked _by_ '.GlobalEnv':
birthwt, housing, quine
The following object is masked from 'package:dplyr':
select
model_negbin <- glm.nb(Days ~ Eth + Sex + Age, data = quine)
model_negbin %>%
glm_coef(labels = model_labels(model_negbin))
```

Parameter | Rate ratio | Pr(>|z|) |

Constant | 20.24 (12.24, 33.47) | < 0.001 |

Ethnicity: White | 0.57 (0.38, 0.84) | 0.005 |

Sex: Male | 1.07 (0.74, 1.53) | 0.73 |

Age group: F1 | 0.69 (0.39, 1.23) | 0.212 |

Age group: F2 | 1.2 (0.7, 2.03) | 0.507 |

Age group: F3 | 1.29 (0.73, 2.28) | 0.385 |

null.deviance | df.null | logLik | AIC | BIC | deviance | df.residual |

192.24 | 145 | -547.83 | 1109.65 | 1130.54 | 167.86 | 140 |

Notice that age group is a factor with more than two levels and is significant:

LR Chisq | Df | Pr(>Chisq) |

12.66 | 1.00 | 0.00 |

0.15 | 1.00 | 0.70 |

9.48 | 3.00 | 0.02 |

Thus, we want to report confidence intervals and \(p\)-values adjusted for multiple comparisons.

Effect plot:

We adjust for multiple comparisons:

contrast | Eth | ratio | SE | z.ratio | p.value | lower.CL | upper.CL |

F0 / F1 | Aboriginal | 1.44 | 0.34 | 1.57 | 0.40 | 0.79 | 2.62 |

F0 / F2 | Aboriginal | 0.84 | 0.19 | -0.77 | 0.86 | 0.46 | 1.51 |

F0 / F3 | Aboriginal | 0.78 | 0.19 | -1.04 | 0.72 | 0.42 | 1.45 |

F1 / F2 | Aboriginal | 0.58 | 0.12 | -2.65 | 0.04 | 0.34 | 0.98 |

F1 / F3 | Aboriginal | 0.54 | 0.12 | -2.89 | 0.02 | 0.31 | 0.93 |

F2 / F3 | Aboriginal | 0.93 | 0.20 | -0.34 | 0.99 | 0.53 | 1.63 |

F0 / F1 | White | 1.44 | 0.34 | 1.57 | 0.40 | 0.79 | 2.62 |

F0 / F2 | White | 0.84 | 0.19 | -0.77 | 0.86 | 0.46 | 1.51 |

F0 / F3 | White | 0.78 | 0.19 | -1.04 | 0.72 | 0.42 | 1.45 |

F1 / F2 | White | 0.58 | 0.12 | -2.65 | 0.04 | 0.34 | 0.98 |

F1 / F3 | White | 0.54 | 0.12 | -2.89 | 0.02 | 0.31 | 0.93 |

F2 / F3 | White | 0.93 | 0.20 | -0.34 | 0.99 | 0.53 | 1.62 |

We can see the comparison graphically with:

We will use an example on the effect of thiotepa versus placebo on the development of bladder cancer.

```
data(bladder)
bladder <- bladder %>%
mutate(times = stop,
rx = factor(rx, labels=c("Placebo", "Thiotepa"))
) %>%
var_labels(times = "Survival time",
rx = "Treatment")
```

Using robust standard errors (default):

Parameter | Survival time ratio | Pr(>|z|) |

Treatment: Thiotepa/Placebo | 1.64 (0.89, 3.04) | 0.116 |

Scale | 1 (0.85, 1.18) | 0.992 |

In this example the scale parameter is not statistically different from one, meaning hazard is constant and thus, we can use the exponential distribution:

```
model_exp <- survreg(Surv(times, event) ~ rx, data = bladder, dist = "exponential")
model_exp %>%
glm_coef(labels = c("Treatment: Thiotepa/Placebo"))
```

Parameter | Survival time ratio | Pr(>|z|) |

Treatment: Thiotepa/Placebo | 1.64 (0.85, 3.16) | 0.139 |

Interpretation:Patients receiving Thiotepa live on average 64% more than those in the Placebo group.

Using naive standard errors:

Parameter | Survival time ratio | Pr(>|z|) |

Treatment: Thiotepa/Placebo | 1.64 (1.11, 2.41) | 0.012 |

Parameter | Hazard ratio | Pr(>|z|) |

Treatment: Thiotepa/Placebo | 0.64 (0.44, 0.94) | 0.02 |

Interpretation:Patients receiving Thiotepa are 64% less likely of dying than those in the Placebo group.

We look at the relationship between sex and age on the distance from the pituitary to the pterygomaxillary fissure (mm).

```
library(nlme)
Attaching package: 'nlme'
The following objects are masked from 'package:ordinal':
VarCorr, ranef
The following object is masked from 'package:dplyr':
collapse
data(Orthodont)
```

We fit the model:

```
model_lme <- lme(distance ~ Sex * I(age - mean(age, na.rm = TRUE)), random = ~ 1|Subject,
method = "ML", data = Orthodont)
```

```
model_lme %>%
glm_coef(labels = c(
"Constant",
"Sex: female-male",
"Age (years)",
"Sex:Age interaction"
))
```

Parameter | Coefficient | Pr(>|t|) |

Constant | 24.97 (24.03, 25.9) | < 0.001 |

Sex: female-male | -2.32 (-3.78, -0.86) | 0.005 |

Age (years) | 0.78 (0.63, 0.94) | < 0.001 |

Sex:Age interaction | -0.3 (-0.54, -0.07) | 0.015 |

Effect plot: