Polynomial Effects

Variation in Infant Mortality Rates Across Countries

Investigating a Linear Effect of Female Education Level

To better understand the causes of variation in infant mortality rates across countries, the number of infant deaths before the age of one per 1000 live births (infant mortality rate) was linearly regressed on the average number of years of formal education for a woman (female education level) for 124 countries ($\mathrm{R}^2 = 0.606$). We controlled for GNI in this regression by including a dummy variable that assigned a 1 to countries with a high-gross national income (GNI) and a 0 to those with a low GNI. The figure and equation below show the model developed from this regression:

Figure 1: The relationship between female education level and fertility rate for women in high and low GNI countries.

\[ \begin{split} \hat{\mathrm{Infant~Mortality~Rate}_i} &= 63.37- 3.6(\mathrm{Female~Education~Level}_i) - 17.58(\mathrm{GNI}_i) \end{split} \]

These results suggest that a one-unit change in female education level (or an additional year of formal education) is associated with a decreased infant mortality rate by 3.60 (or 3.60 less infant deaths before their first birthday, per 1000 live births). It also suggests that countries with a high GNI have a lower infant mortality rate by 17.58.

Investigating a Quadratic Effect of Female Education Level

An investigation of the standardized residuals for this linear model suggest that a linear model may not accurately represent the data and that a model including a quadratic effect of female education may be more appropriate as well as one that segments GNI into four levels. This new model was regressed, controlling for each GNI and was found to better represent the data ($\mathrm{R}^2 = 0.711$) than the linear model ($\mathrm{R}^2 = 0.606$). Again, the equation and a figure for this model are provided below.

Figure 2: Quadratic fits showing the relationship between fertility rate and female education level for four different GNI levels.

\[ \begin{align} \hat{\mathrm{Infant~Mortality~Rate}_i} &= 75.02 - 6.84(\mathrm{Female~Education~Level}_i) + 0.32(\mathrm{Female~Education~Level}_i^2) \nonumber \\ &\quad - 33.37(\mathrm{Upper~GNI}_i) - 24.57(\mathrm{Upper/Middle~GNI}_i) - 9.91(\mathrm{Low/Middle~GNI}_i) \end{align} \]

Like in the linear model, countries with high levels of GNI are associated with a lower infant mortality rate and female education has a negative effect on infant mortality rate, controlling for GNI. However, while the negative effect of female education level remains the same across all levels of female education in the linear model, in the quadratic model, this effect decreases as female education level increases. Another way of saying this is that the negative effect of female education on infant mortality rate is greater at lower levels of education than it is at higher levels (i.e., there is a greater effect at education level 1 than at education level 12). This can be interpreted to mean that the initial years of female education are more critical to decreasing infant mortality than later years. Education still has an effect in these later stages, but its effect diminishes.

--- title: "Polynomial Effects" format: html: fig-width: 12 fig-height: 9 editor: visual execute: echo: false code-tools: true --- ```{r} #| echo: false #| message: false #| warning: false library(tidyverse) library(broom) library(gt) library(ggtext) library(dplyr) fertility <- read_csv("fertility.csv") ``` ## Variation in Infant Mortality Rates Across Countries [CSV](https://raw.githubusercontent.com/zief0002/bespectacled-antelope/main/data/fertility.csv) [Codebook](http://zief0002.github.io/bespectacled-antelope/codebooks/fertility.html) ### Investigating a Linear Effect of Female Education Level To better understand the causes of variation in infant mortality rates across countries, the number of infant deaths before the age of one per 1000 live births (infant mortality rate) was linearly regressed on the average number of years of formal education for a woman (female education level) for 124 countries ($\mathrm{R}^2 = 0.606$). We controlled for GNI in this regression by including a dummy variable that assigned a 1 to countries with a high-gross national income (GNI) and a 0 to those with a low GNI. The figure and equation below show the model developed from this regression: ```{r} #| echo: false #| label: fig-scatterplot #| fig-cap: "The relationship between female education level and fertility rate for women in high and low GNI countries." #| fig-align: right #| fig-width: 10 #| fig-height: 6 #| out-width: 100% #| out-height: 100% #create scatterplot ggplot(fertility, aes(x = educ_female, y = infant_mortality)) + geom_point(alpha = 0.2, size = 2, color = "black", stroke = 0.8) + theme_bw(base_size = 14) + theme( panel.grid = element_blank(), axis.title.x = element_markdown(), axis.title.y = element_markdown(), legend.position = c(0.85, 0.85) ) + geom_abline(aes(intercept = 63.37, slope = -3.6, color = 'Low GNI'), linetype = 'dashed') + geom_abline(aes(intercept = 63.37 - 17.58, slope = -3.6 , color = 'High GNI'), linetype = 'solid') + xlab("Female Eduation Level (years)") + ylab("Infant Mortality Rate (deaths/1000 births)") + scale_color_manual(name = "GNI", values = c("Low GNI" = "blue", "High GNI" = "red"), breaks = c("High GNI", "Low GNI")) + theme( legend.title = element_text(hjust = 0.5)) ``` $$ \begin{split} \hat{\mathrm{Infant~Mortality~Rate}_i} &= 63.37- 3.6(\mathrm{Female~Education~Level}_i) - 17.58(\mathrm{GNI}_i) \end{split} $$ These results suggest that a one-unit change in female education level (or an additional year of formal education) is associated with a decreased infant mortality rate by 3.60 (or 3.60 less infant deaths before their first birthday, per 1000 live births). It also suggests that countries with a high GNI have a lower infant mortality rate by 17.58. ### Investigating a Quadratic Effect of Female Education Level An investigation of the standardized residuals for this linear model suggest that a linear model may not accurately represent the data and that a model including a quadratic effect of female education may be more appropriate as well as one that segments GNI into four levels. This new model was regressed, controlling for each GNI and was found to better represent the data ($\mathrm{R}^2 = 0.711$) than the linear model ($\mathrm{R}^2 = 0.606$). Again, the equation and a figure for this model are provided below. ```{r} fertility <- fertility %>% mutate( gni_class = paste(gni_class, "GNI", sep = " ") ) ``` ```{r} #| echo: false #| label: fig-quadratics #| fig-cap: "Quadratic fits showing the relationship between fertility rate and female education level for four different GNI levels." #| fig-align: left #| fig-width: 10 #| fig-height: 6 #| out-width: 100% #| out-height: 100% #create scatterplot ggplot(fertility, aes(x = educ_female, y = infant_mortality)) + geom_point(alpha = 0.9, aes(fill = gni_class), shape = 21, size = 2, color = "black", stroke = 0.8) + theme_bw(base_size = 14) + theme( panel.grid = element_blank(), axis.title.x = element_markdown(), axis.title.y = element_markdown(), legend.position = c(0.8, 0.8) ) + scale_color_manual(name = "GNI", values = c("Upper GNI" = "red", "Upper/Middle GNI" = "orange", "Low/Middle GNI" = "green", "Low GNI" = "blue"), breaks = c("Upper GNI", "Upper/Middle GNI", "Low/Middle GNI", "Low GNI") ) + scale_fill_manual(name = "GNI", values = c("Upper GNI" = "red", "Upper/Middle GNI" = "orange", "Low/Middle GNI" = "green", "Low GNI" = "blue"), breaks = c("Upper GNI", "Upper/Middle GNI", "Low/Middle GNI", "Low GNI") ) + geom_function(aes(color="Upper GNI"), fun = function(x) {(75.05 - 33.37) - 6.84 * x + 0.32 * x^2} ) + geom_function(aes(color="Upper/Middle GNI"), fun = function(x) {(75.05 - 24.57) - 6.84 * x + 0.32 * x^2} ) + geom_function(aes(color="Low/Middle GNI"), fun = function(x) {(75.05 - 9.91) - 6.84 * x + 0.32 * x^2} ) + geom_function(aes(color="Low GNI"), fun = function(x) {75.05 - 6.84 * x + 0.32 * x^2} ) + xlab("Female Eduation Level (years)") + ylab("Infant Mortality Rate (deaths/1000 live births)") ``` $$ \begin{align} \hat{\mathrm{Infant~Mortality~Rate}_i} &= 75.02 - 6.84(\mathrm{Female~Education~Level}_i) + 0.32(\mathrm{Female~Education~Level}_i^2) \nonumber \\ &\quad - 33.37(\mathrm{Upper~GNI}_i) - 24.57(\mathrm{Upper/Middle~GNI}_i) - 9.91(\mathrm{Low/Middle~GNI}_i) \end{align} $$ Like in the linear model, countries with high levels of GNI are associated with a lower infant mortality rate and female education has a negative effect on infant mortality rate, controlling for GNI. However, while the negative effect of female education level remains the same across all levels of female education in the linear model, in the quadratic model, this effect decreases as female education level increases. Another way of saying this is that the negative effect of female education on infant mortality rate is greater at lower levels of education than it is at higher levels (i.e., there is a greater effect at education level 1 than at education level 12). This can be interpreted to mean that the initial years of female education are more critical to decreasing infant mortality than later years. Education still has an effect in these later stages, but its effect diminishes.