The significance of the sector on the salary of engineers in Sweden

So far I have analysed the effect of experience, education, gender, year and region on the salary of engineers in Sweden. In this post, I will have a look at the effect of the sector on the salary of engineers in Sweden.

Statistics Sweden use NUTS (Nomenclature des Unités Territoriales Statistiques), which is the EU’s hierarchical regional division, to specify the regions.

First, define libraries and functions.

library (tidyverse) 

## -- Attaching packages -------------------------------------- tidyverse 1.3.0 --

## v ggplot2 3.2.1     v purrr   0.3.3
## v tibble  2.1.3     v dplyr   0.8.3
## v tidyr   1.0.2     v stringr 1.4.0
## v readr   1.3.1     v forcats 0.4.0

## -- Conflicts ----------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library (broom) 
library (car)

## Loading required package: carData

## 
## Attaching package: 'car'

## The following object is masked from 'package:dplyr':
## 
##     recode

## The following object is masked from 'package:purrr':
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library (swemaps) # devtools::install_github('reinholdsson/swemaps')
library(sjPlot)

## Registered S3 methods overwritten by 'lme4':
##   method                          from
##   cooks.distance.influence.merMod car 
##   influence.merMod                car 
##   dfbeta.influence.merMod         car 
##   dfbetas.influence.merMod        car

## Install package "strengejacke" from GitHub (`devtools::install_github("strengejacke/strengejacke")`) to load all sj-packages at once!

library(leaps)
library(MASS)

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:dplyr':
## 
##     select

readfile <- function (file1){  
  read_csv (file1, col_types = cols(), locale = readr::locale (encoding = "latin1"), na = c("..", "NA")) %>%  
    gather (starts_with("19"), starts_with("20"), key = "year", value = salary) %>%  
    drop_na() %>%  
    mutate (year_n = parse_number (year))
}
nuts <- read.csv("nuts.csv") %>%
  mutate(NUTS2_sh = substr(NUTS2, 1, 4))
map_ln_n <- map_ln %>%
  mutate(lnkod_n = as.numeric(lnkod)) 

The data table is downloaded from Statistics Sweden. It is saved as a comma-delimited file without heading, 000000CG.csv, http://www.statistikdatabasen.scb.se/pxweb/en/ssd/.

I have renamed the file to 000000CG_sector.csv because the filename 000000CG.csv was used in a previous post.

The table: Average basic salary, monthly salary and women´s salary as a percentage of men´s salary by region, sector, occupational group (SSYK 2012) and sex. Year 2014 - 2018 Monthly salary 1-3 public sector 4-5 private sector

We expect that the sector is an important factor in salaries. As a null hypothesis, we assume that the sector is not related to the salary and examine if we can reject this hypothesis with the data from Statistics Sweden.

tb <- readfile ("000000CG_sector.csv") %>%
  filter (`occuptional  (SSYK 2012)` == "214 Engineering professionals") %>% 
  left_join(nuts %>% distinct (NUTS2_en, NUTS2_sh), by = c("region" = "NUTS2_en")) 

## Warning: Column `region`/`NUTS2_en` joining character vector and factor,
## coercing into character vector

tb_map <- readfile ("000000CG_sector.csv") %>%
  filter (`occuptional  (SSYK 2012)` == "214 Engineering professionals") %>%
  left_join(nuts, by = c("region" = "NUTS2_en")) 

## Warning: Column `region`/`NUTS2_en` joining character vector and factor,
## coercing into character vector

tb_map %>%
  filter (sector == "1-3 public sector") %>%
  right_join(map_ln_n, by = c("Länskod" = "lnkod_n")) %>%
  ggplot() +
    geom_polygon(mapping = aes(x = ggplot_long, y = ggplot_lat, group = lnkod, fill = salary)) +
    facet_grid(. ~ year) + 
    coord_equal() 

tb_map %>%
  filter (sector == "4-5 private sector") %>%
  right_join(map_ln_n, by = c("Länskod" = "lnkod_n")) %>%
  ggplot() +
    geom_polygon(mapping = aes(x = ggplot_long, y = ggplot_lat, group = lnkod, fill = salary)) +
    facet_grid(. ~ year) + 
    coord_equal() 

tb %>%
  ggplot () +  
    geom_point (mapping = aes(x = year_n, y = salary, colour = region, shape=sex)) + 
    facet_grid(. ~ sector) +
  labs(
    x = "Year",
    y = "Salary (SEK/month)"
  )

Before I investigate all possible combinations of the sector and the other factors I shall see if there is some way to predict what factors and interactions that are most significant.

First, use regsubsets to find the model which minimises AIC (Akaike information criterion). Regsubsets is a generic function for regression subset selection with methods for formula and matrix arguments.

b <- regsubsets (log(salary) ~ sector * (year_n + sex + NUTS2_sh), data = tb,  nvmax = 20)
rs <- summary(b)
AIC <- 50 * log (rs$rss / 50) + (2:20) * 2
which.min (AIC)

## [1] 13

names (rs$which[13,])[rs$which[13,]]

##  [1] "(Intercept)"                          
##  [2] "sector4-5 private sector"             
##  [3] "year_n"                               
##  [4] "sexwomen"                             
##  [5] "NUTS2_shSE12"                         
##  [6] "NUTS2_shSE21"                         
##  [7] "NUTS2_shSE22"                         
##  [8] "NUTS2_shSE33"                         
##  [9] "sector4-5 private sector:year_n"      
## [10] "sector4-5 private sector:NUTS2_shSE21"
## [11] "sector4-5 private sector:NUTS2_shSE23"
## [12] "sector4-5 private sector:NUTS2_shSE31"
## [13] "sector4-5 private sector:NUTS2_shSE32"
## [14] "sector4-5 private sector:NUTS2_shSE33"

As a complement, I use stepwise model selection to find the model which fits the data best. StepAIC performs stepwise model selection by AIC.

model <-lm (log(salary) ~ year_n * sex * NUTS2_sh * sector, data = tb) 
b <- stepAIC(model, direction = c("both"))

## Start:  AIC=-1200.79
## log(salary) ~ year_n * sex * NUTS2_sh * sector
## 
##                              Df Sum of Sq      RSS     AIC
## - year_n:sex:NUTS2_sh:sector  7  0.001441 0.041008 -1209.1
## <none>                                    0.039567 -1200.8
## 
## Step:  AIC=-1209.07
## log(salary) ~ year_n + sex + NUTS2_sh + sector + year_n:sex + 
##     year_n:NUTS2_sh + sex:NUTS2_sh + year_n:sector + sex:sector + 
##     NUTS2_sh:sector + year_n:sex:NUTS2_sh + year_n:sex:sector + 
##     year_n:NUTS2_sh:sector + sex:NUTS2_sh:sector
## 
##                              Df Sum of Sq      RSS     AIC
## <none>                                    0.041008 -1209.1
## - year_n:sex:NUTS2_sh         7 0.0047401 0.045748 -1205.6
## - year_n:sex:sector           1 0.0022478 0.043256 -1202.5
## - year_n:NUTS2_sh:sector      7 0.0058131 0.046821 -1201.9
## + year_n:sex:NUTS2_sh:sector  7 0.0014410 0.039567 -1200.8
## - sex:NUTS2_sh:sector         7 0.0080176 0.049026 -1194.5

model <- lm(log(salary) ~ year_n + sex + NUTS2_sh + sector + 
    year_n:sex + year_n:NUTS2_sh + sex:NUTS2_sh + year_n:sector + 
    sex:sector + NUTS2_sh:sector + year_n:sex:NUTS2_sh + year_n:sex:sector + 
    year_n:NUTS2_sh:sector + sex:NUTS2_sh:sector, data = tb)
summary(model)$adj.r.squared 

## [1] 0.9135882

Anova(model, type = 2) %>% 
  tidy() %>% 
  arrange (desc (statistic)) %>% 
  filter(p.value < 0.05) %>% 
  knitr::kable( 
  booktabs = TRUE,
  caption = 'Anova report from linear model fit')

Table 1: Anova report from linear model fit

term	sumsq	df	statistic	p.value
year_n	0.2069351	1	519.760278	0.0000000
sex	0.1113983	2	139.899908	0.0000000
sector	0.0952663	2	119.640560	0.0000000
NUTS2_sh	0.2322097	14	41.660196	0.0000000
year_n:sector	0.0120669	1	30.308411	0.0000003
NUTS2_sh:sector	0.0523275	7	18.775900	0.0000000
year_n:sex	0.0023493	1	5.900761	0.0168659
year_n:sex:sector	0.0022478	1	5.645699	0.0193467
sex:sector	0.0018231	1	4.579079	0.0347260
sex:NUTS2_sh	0.0106289	7	3.813803	0.0010092
sex:NUTS2_sh:sector	0.0080176	7	2.876825	0.0087375
year_n:NUTS2_sh	0.0078670	7	2.822810	0.0098854

There are interactions between the different factors that are significant, i.e. have a p-value less than 0,05 but does not qualify because it´s inclusion in the model does not imply that it lowers the AIC value. The tradeoff between the goodness of fit of the model and the simplicity of the model leads me to exclude those interactions from the model we will examine further.

The model I chose from based on the AIC results is: log(salary) ~ year_n * sector + NUTS2_sh * sector + sex

From this model, the F-value from the Anova table for the sector is 146 (Pr(>F) < 2.2e-16), sufficient for rejecting the null hypothesis that the sector has no effect on the salary holding year as constant. The adjusted R-squared value is 0,870 implying a good fit of the model.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)
tb <- bind_cols(tb, as_tibble(exp(predict(model, tb, interval = "confidence"))))
tb %>%
  ggplot () +  
    geom_point (mapping = aes(x = year_n,y = fit, colour = region, shape=sex)) + 
    facet_grid(. ~ sector) +
  labs(
    x = "Year",
    y = "Salary (SEK/month)"
  )

summary(model) %>%  
  tidy() %>%
  knitr::kable( 
  booktabs = TRUE,
  caption = 'Summary from linear model fit')

Table 2: Summary from linear model fit

term	estimate	std.error	statistic	p.value
(Intercept)	-52.8857464	3.9015473	-13.5550700	0.0000000
year_n	0.0315705	0.0019353	16.3130867	0.0000000
sector4-5 private sector	24.7466021	5.5176204	4.4850135	0.0000150
NUTS2_shSE12	-0.0633886	0.0109476	-5.7901587	0.0000000
NUTS2_shSE21	-0.0951854	0.0109476	-8.6946021	0.0000000
NUTS2_shSE22	-0.0542415	0.0109476	-4.9546264	0.0000020
NUTS2_shSE23	-0.0304669	0.0109476	-2.7829655	0.0061252
NUTS2_shSE31	-0.0213974	0.0109476	-1.9545201	0.0526182
NUTS2_shSE32	-0.0304128	0.0109476	-2.7780207	0.0062142
NUTS2_shSE33	-0.0700399	0.0109476	-6.3977139	0.0000000
sexwomen	-0.0523393	0.0038706	-13.5223569	0.0000000
year_n:sector4-5 private sector	-0.0122815	0.0027369	-4.4873679	0.0000149
sector4-5 private sector:NUTS2_shSE12	0.0069109	0.0154823	0.4463758	0.6560106
sector4-5 private sector:NUTS2_shSE21	-0.0344624	0.0154823	-2.2259214	0.0276066
sector4-5 private sector:NUTS2_shSE22	0.0089387	0.0154823	0.5773509	0.5646232
sector4-5 private sector:NUTS2_shSE23	-0.0206495	0.0154823	-1.3337474	0.1844371
sector4-5 private sector:NUTS2_shSE31	-0.0765503	0.0154823	-4.9443769	0.0000021
sector4-5 private sector:NUTS2_shSE32	-0.0832467	0.0154823	-5.3768944	0.0000003
sector4-5 private sector:NUTS2_shSE33	-0.0711249	0.0154823	-4.5939480	0.0000096

summary(model)$adj.r.squared 

## [1] 0.8699372

Anova(model, type=2) %>% 
  tidy() %>% 
  knitr::kable( 
  booktabs = TRUE,
  caption = 'Anova report from linear model fit')

Table 2: Anova report from linear model fit

term	sumsq	df	statistic	p.value
year_n	0.2069351	1	345.32122	0.00e+00
sector	0.0872899	1	145.66429	0.00e+00
NUTS2_sh	0.1798897	7	42.88421	0.00e+00
sex	0.1095761	1	182.85414	0.00e+00
year_n:sector	0.0120669	1	20.13647	1.49e-05
sector:NUTS2_sh	0.0523275	7	12.47444	0.00e+00
Residuals	0.0844948	141	NA	NA

plot(model, which = 1)

tb[38,]

## # A tibble: 1 x 11
##   region sector `occuptional  (~ sex   year  salary year_n NUTS2_sh    fit
##   <chr>  <chr>  <chr>            <chr> <chr>  <dbl>  <dbl> <chr>     <dbl>
## 1 SE12 ~ 1-3 p~ 214 Engineering~ women 2015   37600   2015 SE12     40664.
## # ... with 2 more variables: lwr <dbl>, upr <dbl>

tb[55,]

## # A tibble: 1 x 11
##   region sector `occuptional  (~ sex   year  salary year_n NUTS2_sh    fit
##   <chr>  <chr>  <chr>            <chr> <chr>  <dbl>  <dbl> <chr>     <dbl>
## 1 SE31 ~ 4-5 p~ 214 Engineering~ men   2015   37900   2015 SE31     41366.
## # ... with 2 more variables: lwr <dbl>, upr <dbl>

tb[76,]

## # A tibble: 1 x 11
##   region sector `occuptional  (~ sex   year  salary year_n NUTS2_sh    fit
##   <chr>  <chr>  <chr>            <chr> <chr>  <dbl>  <dbl> <chr>     <dbl>
## 1 SE21 ~ 4-5 p~ 214 Engineering~ women 2016   34600   2016 SE21     38773.
## # ... with 2 more variables: lwr <dbl>, upr <dbl>

Let’s check what we have found.

For the sake of comparison, a model with no interactions.

model <-lm (log(salary) ~ year_n + sex + NUTS2_sh + sector, data = tb) 
 
plot_model(model, type = "pred", terms = c("NUTS2_sh", "year_n", "sex", "sector"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

## Warning: Package `see` needed to plot multiple panels in one integrated figure.
## Please install it by typing `install.packages("see", dependencies = TRUE)` into
## the console.

## [[1]]

## 
## [[2]]

First, we investigate the interaction between region and sector. All plots below are done with the model which minimised the AIC.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)
 
plot_model(model, type = "pred", terms = c("NUTS2_sh", "sector"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

Also, examine the relationship between gender and sector.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)

plot_model(model, type = "pred", terms = c("sector", "sex"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

And the interaction between year and sector.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)

plot_model(model, type = "pred", terms = c("year_n", "sector"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

The relationship between gender, sector and region.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)

plot_model(model, type = "pred", terms = c("NUTS2_sh", "sector", "sex"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

The relationship between gender, sector and year.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)
 
plot_model(model, type = "pred", terms = c("year_n", "sector", "sex"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

The relationship between region, sector and year.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)
 
plot_model(model, type = "pred", terms = c("NUTS2_sh", "year_n", "sector"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

The relationship between gender, region, sector and year.

model <- model <-lm (log(salary) ~ year_n * sector + NUTS2_sh * sector + sex, data = tb)
 
plot_model(model, type = "pred", terms = c("NUTS2_sh", "year_n", "sector", "sex"))

## Model has log-transformed response. Back-transforming predictions to original response scale. Standard errors are still on the log-scale.

## Warning: Package `see` needed to plot multiple panels in one integrated figure.
## Please install it by typing `install.packages("see", dependencies = TRUE)` into
## the console.

## [[1]]

## 
## [[2]]