1. Data Wrangling in R

  • For today's lab, we return to the CES 2021 data.
library(tidyverse)
library(rio)
library(uwo4419)

# Data
setwd(dirname(rstudioapi::getActiveDocumentContext()$path))
CES_raw <- import("2021 Canadian Election Study v2.0.dta")

## A function to replace -99 by NA
remove99 <- function(x){
  out <- ifelse(x==-99,NA,x)
  return(out)
}

## Cleaning
CES <- factorize(CES_raw) |> 
  # Renaming variables
  rename(UUID=cps21_ResponseId, 
         yob=cps21_yob,
         gender=cps21_genderid,
         prov=cps21_province,
         vote=pes21_votechoice2021,
         pplFeel_Americans=cps21_groups_therm_7,
         pplFeel_Francophones=cps21_groups_therm_3,
         pplFeel_LPC=cps21_party_rating_23,
         pplFeel_CPC=cps21_party_rating_24,
         pplFeel_NDP=cps21_party_rating_25,
         pplFeel_BQ=cps21_party_rating_26,
         pplFeel_GPC=cps21_party_rating_27,
         pplFeel_PPC=cps21_party_rating_29) |> 
  # Making my selection
  select(UUID,yob,gender,prov,vote, 
         pplFeel_Americans,pplFeel_Francophones,
         pplFeel_LPC,pplFeel_CPC,pplFeel_NDP,
         pplFeel_BQ,pplFeel_GPC,pplFeel_PPC) |> 
  # Removing territories
  filter(!(prov %in% c("Northwest Territories","Nunavut","Yukon"))) |> 
  mutate(age=2021-as.numeric(as.character(yob)),
         vt = case_when(
              vote == "Liberal Party" ~ "LPC",
              vote == "Conservative Party" ~ "CPC",
              vote == "ndp" ~ "NDP",
              vote == "Bloc Québécois" ~ "BQ",
              vote == "Green Party" ~ "GPC",
              vote == "People's Party" ~ "PPC",
              vote == "Another party (specify)" ~ "Other",
              vote %in% c("I spoiled my vote","Don't know / Prefer not to answer") ~ NA_character_
            ),
         prov=factor(prov,levels=c("British Columbia","Alberta","Saskatchewan","Manitoba",
                                   "Ontario","Quebec",
                                   "New Brunswick","Prince Edward Island","Nova Scotia",
                                   "Newfoundland and Labrador"))) |> 
  mutate_at(vars(starts_with("pplFeel")),remove99) |> 
  filter(gender %in% c("A man", "A woman")) |> 
  mutate(gender=droplevels(gender))
  • Using the data we just cleaned (CES):
    1. Compute the average score given to each party per province.
    2. Present these averages in a graph where the Y axis are the provinces and the X axis the averages.
CES |> 
  select(-pplFeel_Americans,-pplFeel_Francophones) |> 
  group_by(prov) |> 
  summarise(across(starts_with("pplFeel_"),~mean(.x,na.rm=T))) |> 
  ungroup() |> 
  pivot_longer(-prov,names_pattern = "pplFeel_(.*)",names_to = "Party", values_to = "score") |> 
  mutate(Party=factor(Party,levels=c("LPC","CPC","NDP","BQ","GPC","PPC"))) |> 
  ggplot(aes(x=score,y=prov,color=Party))+
  geom_point() +
  scale_color_manual(values=c("red","blue","orange","cyan","green","purple"))+
  theme_minimal()+
  coord_cartesian(xlim=c(0,100))+
  labs(y="",x="\nFeeling Thermometer Average Score (0-100)")

2. Descriptive Statistics

  • What do these numbers mean?
DAMisc::sumStats(CES,"pplFeel_Francophones") |> 
  arrange(mean)
## # A tibble: 1 × 11
##   variable            mean    sd   iqr   min   q25   q50   q75   max     n   nNA
##   <chr>              <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int> <int>
## 1 pplFeel_Francopho…  67.6  26.5    40     0    50    71    90   100 20797  2251

3. Confidence Intervals

  • There can only be ONE!!! How do we know where our sample stand in the sampling distribution???
  • Confidence intervals: created such that \((1-\alpha)\%\) of the intervals we could make from different samples will cover the true, but unknown population value.
    • Hence, the only thing we can say is that, if \(alpha\) is small enough, we are willing to bet that we are in one of the good samples.
    • Can be computed following \(\bar X\pm z_{crit}\frac{sd_x}{\sqrt{n}}\)

From the CES data:

  1. Compute the average score given to each party per province and their confidence intervals.
  2. Present these in a graph where the Y axis are the provinces and the X axis the averages.
CES |> 
  select(UUID,prov,starts_with("pplFeel_")) |> 
  select(-pplFeel_Americans,-pplFeel_Francophones) |> 
  pivot_longer(c(-UUID,-prov),names_pattern = "pplFeel_(.*)",names_to = "Party", values_to = "score") |> 
  group_by(prov,Party) |> 
  summarise(ci=mean_cl_normal(score)) |> 
  unnest_wider(ci) |> 
  mutate(Party=factor(Party,levels=c("LPC","CPC","NDP","BQ","GPC","PPC"))) |> 
  ggplot(aes(x=prov,y=y,ymin=ymin,ymax=ymax,color=Party))+
  annotate('rect', xmin = seq(.5,8.5,2), xmax = seq(1.5,9.5,2), 
  ymin=-Inf, ymax=Inf, alpha=0.08, fill="black")+
  geom_pointrange(size=0.1,position=position_dodge(0.5)) +
  scale_color_manual(values=c("red","blue","orange","cyan","green","purple"))+
  theme_minimal()+
  theme(legend.position = "top")+
  coord_flip(ylim=c(0,100))+
  labs(x="",y="\nFeeling Thermometer Average Score (0-100)")+
  guides(color = guide_legend(nrow = 1))

4. Hypothesis Testing

  • We know from the Weak Law of Large Numbers that our sample mean converges to the population mean as \(n\) increases.
  • We know from the Central Limit Theorem that our sampling distribution will be Normal (or Student's t for small samples).
  • Setting our null hypothesis as \(\mu_0=0\), we know that either: \[\bar Y \sim N(\mu,\frac{sd_Y}{\sqrt{n}})~~\text{or}~~\bar Y \sim t_{n-1}(\mu,\frac{sd_Y}{\sqrt{n}})\]
  • Therefore, we can compute \(z\) or \(t\) scores depending on our sample size: \[z=\frac{\bar Y-\mu_0}{\sigma}~~\text{or}~~t=\frac{\bar Y-\mu_0}{\frac{sd_Y}{\sqrt{n}}}\]
    • Essentially how many standard deviations below or above \(\mu_0\) an observation \(\bar Y\) is.
    • We use the \(z\) or \(t\) scores to compute our p-values.

  • For proportion, we use the \(z\) distribution where \(se=sd_Y=\sqrt{\frac{p(1-p)}{n}}\).
    • But, we need \(np\geq5\) AND \(n(1-p)\geq5\) for the normal approximation to work.
    • Where \(n\) is the sample size and \(p\) is the hypothesized population proportion (our \(\mu_0\) from above).
  • For difference of means:
    • Usually we have \(H_0: \mu_0=0\).
    • For two means we have \(H_0=\mu_1=\mu_2\) or \(H_0=\mu_1-\mu_2=0\), hence:
      • \(H_A:\mu_1\neq\mu_2 \rightarrow H_A:\mu_1-\mu_2\neq0\)
      • \(H_A:\mu_1<\mu_2 \rightarrow H_A:\mu_1-\mu_2<0\)
      • \(H_A:\mu_1>\mu_2 \rightarrow H_A:\mu_1-\mu_2>0\)
    • So, our \(z-\) or \(t-\)statistic: \(\frac{Estimate~-~H_0Value}{SE}\), with \(SE=\sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}\).
    • Paired t-tests imply that you have the same observations but different variables for both means.
  • A few interpretation exercises:
  1. Is there a difference in the average score given to the CPC between men and women?
tTest("gender","pplFeel_CPC",data=CES)
## Summary:
##            mean     n     se      
## A man      46.20117 9380  31.91189
## A woman    38.7022  11172 30.65365
## Difference 7.498971 20552 0.438948
## 
## 
##  Welch Two Sample t-test
## 
## data:  pplFeel_CPC by gender
## t = 17.084, df = 19642, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group A man and group A woman is not equal to 0
## 95 percent confidence interval:
##  6.638596 8.359346
## sample estimates:
##   mean in group A man mean in group A woman 
##              46.20117              38.70220
  1. Are liberals ranked higher than conservatives on average?
t.test(CES$pplFeel_LPC,CES$pplFeel_CPC,paired = T,alternative = "greater")
## 
##  Paired t-test
## 
## data:  CES$pplFeel_LPC and CES$pplFeel_CPC
## t = 14.114, df = 20459, p-value < 2.2e-16
## alternative hypothesis: true mean difference is greater than 0
## 95 percent confidence interval:
##  4.562733      Inf
## sample estimates:
## mean difference 
##        5.164663

5. Association Tests

Test Data type Why do we use it? Interpretation
\(\chi^2\) Nominal Are the frequencies of the contingency table of the two variables randomly distributed? \(H_0\): Frequencies are randomly distributed.
\(\phi\) Nominal
(2x2 table)		 How strong is the relationship (-1 to 1)? 0=no
>0.1=moderate
>0.3=strong
Cramer's V Nominal
(>2x2 table)		 How strong is the relationship (-1 to 1)? 0=no
>0.1=moderate
>0.3=strong
\(\lambda\) Nominal What is the exact improvement in error in predicting DV by knowing IV? Percent improvement
\(\gamma\) Ordinal How strong is the relationship (-1 to 1)? 0=no
>0.1=moderate
>0.3=strong
Somer's d Ordinal How strong is the relationship (-1 to 1)?
(A \(\gamma\) that accounts for ties on IV)		 0=no
>0.1=moderate
>0.3=strong
Kendall's \(\tau_b\) Ordinal How strong is the relationship (-1 to 1)?
(A \(\gamma\) that accounts for ties on IV and DV)		 0=no
>0.1=moderate
>0.3=strong
Spearman's \(\rho_s\) Ordinal How strong is the relationship (-1 to 1)?
(Talk of a correlation. Testing for monotonic relationship)		 0=no
>0.1=moderate
>0.3=strong
Pearson r Continuous How strong is the relationship (-1 to 1)?
(Talk of a correlation. Testing for linear relationship)		 0=no
>0.1=moderate
>0.3=strong
  1. Is there a relationship between vote and pplFeel_Francophones?
makeStats(CES$vote,CES$pplFeel_Francophones,lambda=T,cramersV=T,chisq=T,gamma=T,d=T,taub=T,rho=T)
##                       statistic p-value
## Chi-squared           2202.8482       0
## Cramers V                0.1513       0
## Lambda                   0.0551       0
## Kruskal-Goodman Gamma    0.0602       0
## Somers D                 0.0514       0
## Tau-b                    0.0517       0
## Spearmans Rho            0.0707       0
  1. Is there a linear relationship between pplFeel_Francophones and feelings towards the parties?
DAMisc::pwCorrMat(pplFeel_Francophones~pplFeel_LPC+pplFeel_CPC+
                    pplFeel_NDP+pplFeel_BQ+pplFeel_GPC+pplFeel_PPC,data=CES)
## Pairwise Correlations
##                      pplFeel_Francophones pplFeel_LPC pplFeel_CPC pplFeel_NDP
## pplFeel_Francophones                                                         
## pplFeel_LPC          0.162*                                                  
## pplFeel_CPC          -0.112*              -0.356*                            
## pplFeel_NDP          0.108*               0.419*      -0.302*                
## pplFeel_BQ           0.398*               0.020*      0.049*      0.068*     
## pplFeel_GPC          0.092*               0.278*      -0.165*     0.528*     
## pplFeel_PPC          -0.140*              -0.217*     0.302*      -0.134*    
##                      pplFeel_BQ pplFeel_GPC pplFeel_PPC
## pplFeel_Francophones                                   
## pplFeel_LPC                                            
## pplFeel_CPC                                            
## pplFeel_NDP                                            
## pplFeel_BQ                                             
## pplFeel_GPC          0.134*                            
## pplFeel_PPC          0.044*     0.109*