```
# Loading the packages
knitr::opts_chunk$set(warning = FALSE, message = FALSE)
library(broom)
library(openintro)
library(tidyverse)
library(cesR)
library(knitr)
library(tibble)
library(ggplot2)
library(ggpubr)
library(flextable)
library(car)
```

## Introduction

After experiencing COVID-19 and continued economic uncertainty in Canada, it is crucial for Canadians to consider how the 2025 federal election will affect their life in Canada and how each party will put Canada on the road to economic recovery[21]. The liberal party has won the 2021 federal election, and the conservative party has the second highest votes[21], so we want to anticipate whether the Liberals maintain their winning streak or the Conservatives come from behind to defeat the Liberals in the 2025 federal election.

To help the public have a clear insight into the vote of two parties in the 2025 federal election, we first develop a research question: ’ which of the two political parties (Liberal and Conservative) will win the election based on people’s demographics in Canada? , then we constructed the statistical models based on the survey dataset about the previous federal election. Eventually, after analyzing the probability of voting for a particular party by dividing the entire population into various cells, we can anticipate the 2025 federal election vote using the entire population, which is what we call post-stratification in statistics.

The two modelling datasets are the 2019 Canadian Election Study Phone Survey dataset[5] and the 2017 census dataset “General Social Survey on Social Identity.”[4] The 2017 General Social Survey on Social Identity dataset indicates the demographics of people in Canada. It can be used to gain inference on statistics of the different kinds of people living around Canada. The “2019 Canadian Election study Phone Survey attitudes and opinions of Canadians for the 2019 federal election.

### Hightlight of Hypothesis

Since the Liberal party has won the 2021 federal election, it is reasonable for us to devised the hypothesis that the Liberal party will win the 2025 election Specifically, our hypothesis is that the vote of liberal party will be the higher than others in 2025 election after conducting statistical prediction.

### Terminologies introduction

```
# Generate the table view of important terminologies
all_terms <- data.frame(Term = c("Liberal", "Conservatives", "Census data", "Variable"),
Description = c("Grits, Canada Liberal Party",
"Tory, PCs, Conservative Party of Canada",
"Data that studies (social survey) a number of the population",
"Each variable contains fluctuating value depends on its respondent) (e.g. sex, income, education degree)"
))
knitr::kable(all_terms, caption = "Terminologies")
```

Term | Description |
---|---|

Liberal | Grits, Canada Liberal Party |

Conservatives | Tory, PCs, Conservative Party of Canada |

Census data | Data that studies (social survey) a number of the population |

Variable | Each variable contains fluctuating value depends on its respondent) (e.g. sex, income, education degree) |

Terminologies

In general, we provided a detailed description of the datasets as well as data preparation (i.e data cleaning)for future modelling in the section data. We also gave a comprehensive introduction of the methods we used for the modelling and poststratification in the methods part. The results of the model analysis and poststratification prediction were provided in the results part. Ultimately, we considered the limitation of our analysis and anticipation

## Data

```
# import dataset
census_data <-read_csv("gss_clean.csv")
survey_data <- read_csv("ces2019-phone_clean.csv")
# see the data
head(census_data)
```

```
## # A tibble: 6 × 81
## caseid age age_first_child age_youngest_child_under_6 total_children
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 52.7 27 NA 1
## 2 2 51.1 33 NA 5
## 3 3 63.6 40 NA 5
## 4 4 80 56 NA 1
## 5 5 28 NA NA 0
## 6 6 63 37 NA 2
## # ℹ 76 more variables: age_start_relationship <dbl>,
## # age_at_first_marriage <dbl>, age_at_first_birth <dbl>,
## # distance_between_houses <dbl>, age_youngest_child_returned_work <dbl>,
## # feelings_life <dbl>, sex <chr>, place_birth_canada <chr>,
## # place_birth_father <chr>, place_birth_mother <chr>,
## # place_birth_macro_region <chr>, place_birth_province <chr>,
## # year_arrived_canada <chr>, province <chr>, region <chr>, …
```

```
head(survey_data)
```

```
## # A tibble: 6 × 278
## sample_id survey_end_CES survey_end_month_CES survey_end_day_CES
## <dbl> <dttm> <dbl> <dbl>
## 1 18 2019-09-23 21:48:29 9 23
## 2 32 2019-09-13 00:02:30 9 12
## 3 39 2019-09-11 00:02:33 9 10
## 4 59 2019-10-10 21:20:13 10 10
## 5 61 2019-09-12 22:28:58 9 12
## 6 69 2019-09-17 23:56:26 9 17
## # ℹ 274 more variables: num_attempts_CES <dbl>, interviewer_id_CES <dbl>,
## # interviewer_gender_CES <chr>, language_CES <dbl>, phonetype_CES <dbl>,
## # survey_end_PES <dttm>, survey_end_month_PES <dbl>,
## # survey_end_day_PES <dbl>, num_attempts_PES <dbl>, interviewer_id_PES <dbl>,
## # interviewer_gender_PES <chr>, language_PES <dbl>, phonetype_PES <dbl>,
## # mode_PES <dbl>, phone_type <dbl>, weight_CES <dbl>, weight_PES <dbl>,
## # c1 <dbl>, c2a <dbl>, c3 <dbl>, q1 <dbl>, q2 <dbl>, q3 <dbl>, q4 <dbl>, …
```

### Dataset

The dataset “Canadian Election Study 2019, Phone Survey”[5] was used for modelling and analysis. The 2019 Canadian Election Study was conducted to gather Canadians’ attitudes, opinions and characteristics during and after the 2019 federal election. This data set recorded the interviews’ responses using phone interviews, and it included 4,021 cases and took survey time, interviewer information and question numbers as column names. This survey contains 2 components the data set containing the question numbers and responses; the other is the survey document with text questions and answer options.

The dataset “Canadian General Social Survey,”[4] which Statistics Canada conducted in 2017, was used to anticipate how the entire population voted in the 2025 federal election. This was census data conducted by telephone across ten provinces. This data set provided respondents’ information and household information, such as age, sex, education, income, household income, etc. (For the detailed variables, see Appendix). Since we need to analyze the model and predict the two parties’ votes based on these two data sets, matching the variables between survey data and census data was vital and it was described in the data cleaning.

### Data Cleaning

#### Survey data cleaning

Find the interest of variables and rename these variables into an understandable format(See important variables table below)

In order to match the variables and values in the census data, which will be helpful for our further analysis, we filtered the step to clean the survey data:

- We filtered out the value “Other” under the gender variable. Since there was only one case from 4,021 cases, it would not affect the overall results.
- We only wanted to study two major parties that are liberal and conservative. Thus, for the variable “Which party you will likely vote for”, we filtered out the cases in that people responded “Don’t know/ Undecided” and “Refused”. When people who choose these responses choose a party to vote for, it also affects the total number of votes. So these are also indeterminate votes.
- For the variable”Household Income”, We filtered out the cases in which people responded “Don’t know” and “Refused”.
- Lastly, in case there is some error when recording the household size response, we only make a limit to choosing households size greater than 1.

In order to figure out the proportion of liberal and conservative votes in the survey data, we created two new binary variables “vote for liberal,” which holds a value of 1 if people chose to vote Liberal Party, 0 otherwise; “vote for conservative” which contains a value 1 if people decided to vote Conservative Party, 0 otherwise.

As mentioned above, we needed to match the variables and values that each variable hold between survey data and census data. (1)We divided “household income” into different groups, which are “Less than $25,000”, “$25,000 to $49,999”,“$50,000 to $74,999”, “$75,000 to $99,999”,“$100,000 to $ 124,999” and “$125,000 and more”. (2)We put “respondents’ birthplace” into three categories which are “Born in Canada”, “Born outside Canada”, and “Don’t know”.(3)We put “respondents educational degrees” into 6 categories which are corresponded to the categories in census data. The 6 categories are “Less than high school diploma or its equivalent”, “High school diploma or a high school equivalency certificate”, “College, CEGEP or other non-university certificate or di…”, “University certificate or diploma below the bachelor’s level”, “Bachelor’s degree (e.g. B.A., B.Sc., LL.B.)”, “University certificate, diploma or degree above the bachelor’s degree…”.

#### Census dataset cleaning

Because there was no category in the survey data which was corresponded to “Trade certificate or diploma” in the census data. We filtered out the education degree which was “Trade certificate or diploma”.

Since only Canadian residents and permanent residents who are at least 18 years old can vote. We filtered out respondents who are under 18 years old in both data sets.

Eventually, in order to make the dataset tidy and easy to read, we only select the columns that we am interested.

```
### Data Cleaning Survey data
survey_data <-
survey_data %>%
filter(q3 != 3, q69 != -8, q69 != -9, q11 != -8, q11 != -9, q11 != -7) %>%
mutate(age = 2019-q2,
sex = case_when(q3 == 1 ~ "Male",
q3 == 2 ~ "Female"),
vote_liberal = ifelse(q11==1, 1, 0),
vote_cons = ifelse(q11== 2, 1, 0),
vote_ndp = ifelse(q11==3, 1, 0),
place_birth_canada = case_when(q64 %in% c(1,2) ~ "Born in Canada",
q64 %in% c(3:13) ~ "Born outside Canada",
q64 %in% c(-8,-9) ~ "Don't know"),
hh_size = q71,
income_family = case_when(q69 < 25000 ~ "Less than $25,000",
25000 <= q69 & q69 <= 49999 ~"$25,000 to $49,999",
50000 <= q69 & q69 <= 74999 ~"$50,000 to $74,999",
75000 <= q69 & q69 <= 99999 ~"$75,000 to $99,999",
100000 <= q69 & q69 <= 124999 ~"$100,000 to $ 124,999",
q69 >= 125000 ~"$125,000 and more"),
education = case_when(q61 %in% c(1:4)~"Less than high school diploma or its equivalent",
q61 %in% c(5)~"High school diploma or a high school equivalency certificate",
q61 %in% c(6,7)~"College, CEGEP or other non-university certificate or di...",
q61 %in% c(8)~"University certificate or diploma below the bachelor's level",
q61 %in% c(9)~"Bachelor's degree (e.g. B.A., B.Sc., LL.B.)",
q61 %in% c(10,11)~"University certificate, diploma or degree above the bach..."),
province = case_when(q4 == 1 ~ "Newfoundland and Labrador",
q4 == 2 ~ "Prince Edward Island",
q4 == 3 ~ "Nova Scotia",
q4 == 4 ~ "New Brunswick",
q4 == 5 ~ "Quebec",
q4 == 6 ~ "Ontario",
q4 == 7 ~ "Manitoba",
q4 == 8 ~ "Saskatchewan",
q4 == 9 ~ "Alberta",
q4 == 10 ~ "British Columbia"))%>%
select(age,sex,vote_liberal,vote_cons,vote_ndp,place_birth_canada,hh_size,
income_family,education,province)%>%filter(hh_size >= 1, age >= 18)
# remove the negative value of hh size and people who are at least 18 years old
# Remove NA
survey_data <- na.omit(survey_data)
survey_data <- survey_data %>% mutate(age_group = case_when(age<25 ~ "Youth", age>= 25 & age <65 ~"Adults", age>= 65 ~"Seniors"))
# see the cleaned version
head(survey_data)
```

```
## # A tibble: 6 × 11
## age sex vote_liberal vote_cons vote_ndp place_birth_canada hh_size
## <dbl> <chr> <dbl> <dbl> <dbl> <chr> <dbl>
## 1 25 Male 1 0 0 Born in Canada 1
## 2 19 Male 0 0 0 Born in Canada 5
## 3 35 Male 0 0 1 Born in Canada 1
## 4 80 Male 0 0 0 Born in Canada 1
## 5 20 Male 0 0 0 Born in Canada 1
## 6 24 Male 0 0 0 Born in Canada 2
## # ℹ 4 more variables: income_family <chr>, education <chr>, province <chr>,
## # age_group <chr>
```

```
# 2017 census data(GSS), age in 2019 = age in 2017 + 2
census_data <- census_data %>%
mutate(age=round(age)+2) %>%
select(age,sex,place_birth_canada,hh_size,
province,income_family,education) %>%
filter(education != "Trade certificate or diploma")
# Remove NA from census data
census_data <- na.omit(census_data)
# See the cleaned data
head(census_data)
```

```
## # A tibble: 6 × 7
## age sex place_birth_canada hh_size province income_family education
## <dbl> <chr> <chr> <dbl> <chr> <chr> <chr>
## 1 55 Female Born in Canada 1 Quebec $25,000 to $49,… High sch…
## 2 66 Female Born in Canada 2 Ontario $75,000 to $99,… Bachelor…
## 3 82 Female Born in Canada 2 Alberta $100,000 to $ 1… High sch…
## 4 30 Male Born in Canada 2 Quebec $50,000 to $74,… College,…
## 5 65 Female Born in Canada 2 Quebec $50,000 to $74,… High sch…
## 6 61 Female Born in Canada 1 Nova Scotia Less than $25,0… Less tha…
```

### Important variables

```
all_variables <- data.frame(Variable = c("q2","q3", "q11", "q11", "q64", "q71", "q69", "q61", "q4"),
Cleaned_Variable = c("age", "sex", "vote_liberal", "vote_cons", "place_birth", "hh_size", "income_family", "education", "province"),
Type = c("Numerical", "Categorical", "Categorical", "Categorical", "Categorical", "numerical", "Numerical", "Categorical", "Categorical"),
Description = c("Numerical value of repondent's age", "Sex of repondent", "If the respondent voted liberal party, then 1. Otherwise, 0", "If the respondent voted conservative party, then 1. Otherwise, 0", "This variable holds three categories which are Born in Canada, Born outside Canada, and Don't know", "Household size", "Categorized income of the respondent", "Respondent's education degree", "Province of the respondent is currently living")
)
names(all_variables)[2] <- "Cleaned Variable"
knitr::kable(all_variables, caption = "Important variables")
```

Variable | Cleaned Variable | Type | Description |
---|---|---|---|

q2 | age | Numerical | Numerical value of repondent’s age |

q3 | sex | Categorical | Sex of repondent |

q11 | vote_liberal | Categorical | If the respondent voted liberal party, then 1. Otherwise, 0 |

q11 | vote_cons | Categorical | If the respondent voted conservative party, then 1. Otherwise, 0 |

q64 | place_birth | Categorical | This variable holds three categories which are Born in Canada, Born outside Canada, and Don’t know |

q71 | hh_size | numerical | Household size |

q69 | income_family | Numerical | Categorized income of the respondent |

q61 | education | Categorical | Respondent’s education degree |

q4 | province | Categorical | Province of the respondent is currently living |

Important variables

## Explotory Data Analysis

### Numerical Summary Table

```
# average age
#table
table1 <- survey_data %>% group_by(province) %>%
filter(vote_liberal==1) %>% summarise(average_age = sum(age)/n(), sd=sd(age))
table2 <- survey_data %>% group_by(province) %>%
filter(vote_cons==1) %>% summarise(average_age = sum(age)/n(), sd= sd(age))
mean_sd <- data.frame(Province = table1$province, table1$average_age, table1$sd, table2$average_age, table2$sd)
names(mean_sd)[2] <- 'Average Age(Liberal Party)'
names(mean_sd)[4] <- 'Average Age(Conservative Party)'
names(mean_sd)[3] <- 'Standard Deviation(Liberal Party)'
names(mean_sd)[5] <- 'Standard Deviation(Conservative Party)'
knitr::kable(mean_sd, caption = "Average age of respondents who voted Liberal Party or Conservative Party")
```

Province | Average Age(Liberal Party) | Standard Deviation(Liberal Party) | Average Age(Conservative Party) | Standard Deviation(Conservative Party) |
---|---|---|---|---|

Alberta | 45.50000 | 17.56823 | 47.94545 | 15.23687 |

British Columbia | 52.50000 | 18.30700 | 51.92969 | 16.12363 |

Manitoba | 54.73333 | 17.72056 | 53.43038 | 15.32359 |

New Brunswick | 52.32432 | 13.99812 | 54.00000 | 16.14001 |

Newfoundland and Labrador | 54.76190 | 15.20881 | 53.20000 | 13.76252 |

Nova Scotia | 52.06818 | 17.57958 | 53.08824 | 14.56098 |

Ontario | 53.29703 | 16.25759 | 54.88435 | 16.21876 |

Prince Edward Island | 59.04444 | 15.16119 | 55.36364 | 15.21680 |

Quebec | 48.45270 | 15.16231 | 47.98551 | 14.83189 |

Saskatchewan | 51.81818 | 19.47070 | 48.77228 | 16.40846 |

Average age of respondents who voted Liberal Party or Conservative Party

**Description:** The average age and standard deviation of Liberal Party
and Conservative Party voters are displayed in the table below. As can
be seen, there is not a significant age gap between the voters of these
two parties in each province.

```
#pie chart voters based on sex
plot_da1 <- survey_data %>% filter(vote_cons == 1)
plot_da2 <- survey_data %>% filter(vote_liberal == 1)
a<-ggplot(plot_da2, aes(x="sex", y=vote_liberal, fill=sex))+
labs(title="The percentage of male and female Liberal Party voters") +
theme(plot.title = element_text(hjust = 0))+
geom_bar(width = 1, stat = "identity") +
coord_polar("y", start=0) + theme_void()+
theme(plot.title = element_text(hjust = 0.5))
b<-ggplot(plot_da1, aes(x="sex", y=vote_cons, fill=sex))+
labs(title=" The percentage of male and female Conservative Party voters" ) +
theme(plot.title = element_text(hjust = 0))+
geom_bar(width = 1, stat = "identity") +
coord_polar("y", start=0) + theme_void()+
theme(plot.title = element_text(hjust = 0.5))
ggarrange(a,b,nrow = 2,ncol = 1)
```

**Description:** This pie chart shows the percentage of male and female
liberal and conservative party voters separately. From the picture, we
can see that among the conservative party voters, the percentage of male
voters is much higher than the female voters. Among the liberal party
voters, there was less different among the sex of the voters.

```
# education and vote
ggplot(data = survey_data, aes(x=education, group= vote_liberal, fill=factor(vote_liberal,
labels = c("not vote for the Liberal Party",
"vote for the Liberal Party")))) +
geom_bar(position = position_dodge(0.9, preserve = "single")) +
labs(title = "The distribution of Liberal Party voters
based on education degree",
x="Education Degree", y="Number of people", fill= "Vote Liberal") +
scale_fill_brewer(palette = "Dark2") +
theme_light() + coord_flip()+theme(plot.title = element_text(hjust = 0))
```

**Description:** The plot shows the education demographic for Liberal
Party voters and non Liberal Party voters. From the plot, we can see
respondents with College, CEGEP or other non-university certificate took
the highest proportion on the non Liberal Party voters. Respondents with
bachelor degree took the highest percentage on the total number of votes
for Liberal Party.

```
ggplot(data = survey_data, aes(x=education, group= vote_cons, fill=factor(vote_cons, labels = c("not vote for the Conservative Party",
"vote for the Conservative Party")))) +
geom_bar(position = position_dodge(0.9, preserve = "single")) +
labs(title = "The distribution of Conservative Party voters
based on education degree",
x="Education Degree", y="Number of people", fill= "Vote Conservative") +
scale_fill_brewer(palette = "Dark2") +
theme_light()+coord_flip()+theme(plot.title = element_text(hjust = 0))
```

**Description:** The plot shows the education demographic for
Conservative Party voters and non Conservative Party voters. From the
plot, we can see respondents with bachelors degree took the highest
percentage on the non Conservative Party voters. Respondents with
College, CEGEP or other non-university certificate took the highest
percentage on the Conservative Party Voters.The educational backgrounds
of the voters and non-voters of the two parties separately are opposite.

```
#income and vote
plot5 <- ggplot(data = survey_data, aes(x=income_family, group= vote_liberal, fill=factor(vote_liberal,
labels = c("not vote for the Liberal Party",
"vote for the Liberal Party")))) +
geom_bar(position = position_dodge(0.9, preserve = "single")) +
labs(title = "The distribution of Liberal Party voters
based on household income",
x="Household Income", y="Number of people", fill= "Vote Liberal") +
scale_fill_brewer(palette = "Dark2") +
theme_light() + coord_flip()
plot5 <- plot5 + theme(plot.title = element_text(hjust = 0))
plot6 <- ggplot(data = survey_data, aes(x=income_family, group= vote_cons, fill=factor(vote_cons, labels = c("not vote for the Conservative Party",
"vote for the Conservative Party")))) +
geom_bar(position = position_dodge(0.9, preserve = "single")) +
labs(title = "The distribution of Conservative Party voters
based on household income",
x="Household Income", y="Number of people", fill= "Vote Conservative") +
scale_fill_brewer(palette = "Dark2") +
theme_light() + coord_flip()
plot6 <- plot6 + theme(plot.title = element_text(hjust = 0))
figure_in <- ggarrange(plot5, plot6,ncol = 1, nrow = 2)
figure_in
```

**Description:** The above plot represents the household income
demographic for Liberal Party and non-Liberal Party voters. According to
the plot, the group “$125,000 and more” took the highest proportion of
non-Liberal Party voters. However, the group “125,000 and more” took the
highest proportion of those who will vote for the Liberal Party.

The below plot represents the household income demographic for Conservative Party and non-Conservative Party voters. According to the plot, the group “125,000 and more” took the highest proportion of non-Conservative Party voters. However, the group “125,000 and more” took the highest proportion of those who will vote for the Conservative Party.

## Methods

### Methodology introduction

This analysis aims to predict the overall popular vote of the next Canadian federal election in 2025(tentatively) using a regression model[6] with post-stratiﬁcation[9]. To achieve the goal, we will build the logistic regression models with all predictors first and then perform the stepwise model selection by AIC[] to confirm the final model with specific predictors. The reason why we use logistic regression model is that the outcome variable in the model is binary, which simply only has 0 represents for not vote for liberal and 1 represents for vote for liberal[23].

The stepwise variable selection essentially adds or deletes variables based on which model has the smallest AIC after each addition/deletion. The methods will add/delete a predictor based on trying to add/delete every available predictor at a time, then choose to keep the model results with the smallest AIC value compared to the model we start with. Moreover, as forward and backward selection only explores the portion of all possible models, we use the stepwise selection to account for this dependence. The stepwise selection process will iterate between forward and backwards selection until we can not add or delete the predictors anymore[23].

### Model Specifics

#### Model for predicting the liberal party

We would be using a logistic regression model to predict the proportion of voters who will vote for the liberal party. The model from variable selection comes with five predictors ,including numerical variable age, four categorical variables birth place in Canada, province,education degree, and sex.However, for better comparison and anticipation between conservatives party and Liberal party, we decided to add the income family in the final model. Here is the final logistic model with six predictors we would be using to predict the proportion of voters who will vote for the liberal party.

- Where
*p*represents the probability of voting for the liberal party *β*_{0}represents the intercept of the model*β*_{1}represents A one year increase in age is associated with a*β*_{1}increase in the log-odds of voting for liberal party.*β*_{2}represents the average difference in log odds of voting for liberal party between male and female for a certain age, birth place Canada, province and education.*β*_{3}represents the average difference in log odds of voting for liberal party between people born outside Canada and those who born in Canada(baseline) and ,holding other constant*β*_{4}represents the average difference in log odds of voting for liberal party between people don’t know if they born in Canada or not and those who born in Canada(baseline) and ,holding other constant*β*_{5}represents the the average difference in log odds of voting for liberal party between people live in British Columbia and people live in Alberta(baseline), holding other constant.*β*_{6}represents the the average difference in log odds of voting for liberal party between people live in Manitoba and people live in Alberta(baseline), holding other constant.*β*_{7}represents the the average difference in log odds of voting for liberal party between people live in New Brunswick and people live in Alberta(baseline), holding other constant.*β*_{8}represents the the average difference in log odds of voting for liberal party between people live in Newfoundland and Labrador and people live in Alberta(baseline), holding other constant.*β*_{9}represents the the average difference in log odds of voting for liberal party between people live in Nova Scotia and people live in Alberta(baseline), holding other constant.*β*_{10}represents the the average difference in log odds of voting for liberal party between people live in Ontario and people live in Alberta(baseline), holding other constant.*β*_{11}represents the the average difference in log odds of voting for liberal party between people live in Prince Edward Island and people live in Alberta(baseline), holding other constant.*β*_{12}represents the the average difference in log odds of voting for liberal party between people live in Quebec and people live in Alberta(baseline), holding other constant.*β*_{13}represents the the average difference in log odds of voting for liberal party between people live in Saskatchewan and people live in Alberta(baseline), holding other constant.*β*_{14}represents the the average difference in log odds of voting for liberal party between people with non-university certificate and people with bachelor’s degree(baseline), holding other constant*β*_{15}represents the the average difference in log odds of voting for liberal party between people with High school diploma and people with bachelor’s degree(baseline), holding other constant*β*_{16}represents the the average difference in log odds of voting for liberal party between people with non-university certificate and people with bachelor’s degree(baseline), holding other constant*β*_{17}represents the the average difference in log odds of voting for liberal party between people with below the bachelor’s and people with bachelor’s degree(baseline), holding other constant*β*_{18}represents the the average difference in log odds of voting for liberal party between people with degree above the bachelor’s and people with bachelor’s degree(baseline), holding other constant*β*_{19}represents the the average difference in log odds of voting for liberal party between income =125,000 and more and income= 100,000 to 124,999(baseline), holding other constant.*β*_{20}represents the the average difference in log odds of voting for liberal party between income =25,000 to 49,999 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{21}represents the the average difference in log odds of voting for liberal party between income =50,000 to 74,999 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{22}represents the the average difference in log odds of voting for liberal party between income =75,000 to 99,999 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{23}represents the the average difference in log odds of voting for liberal party between income =Less than 25,000 and income= 100,000 to 124,999(baseline), holding other constant.

```
# Liberal GLM fit
full_model <- glm(vote_liberal~age+sex+place_birth_canada+hh_size+province+
income_family+education, data= survey_data,family="binomial")
summary(full_model)
# Stepwise Selection based on AIC
sel.var.aic1 <- step(full_model, trace = 0, k = 2, direction = "both")
sel.var.aic1<-attr(terms(sel.var.aic1), "term.labels")
# selected model
model_liberal <- glm(vote_liberal~ age + sex + place_birth_canada + province+
education + income_family,
data= survey_data,family="binomial")
summary(model_liberal)
```

#### Model for predicting the conservatives party

We would be using a logistic regression model[6] to predict the proportion of voters who will vote for the conservatives party. This is the final model with seven predictors after the variable selection,including numerical variable age and household size, five categorical variables birth place in Canada, province,education degree, sex and family income. However, for better comparison and anticipation between conservatives party and Liberal party, we decided to delete the household size in the final model. Here is the final logistic model with six predictors we would be using to predict the proportion of voters who will vote for the conservatives party.

- Where
*p*represents the probability of voting for the conservatives party *β*_{0}represents the intercept of the model*β*_{1}represents on average, a one year increase in age is associated with a*β*_{1}increase in the log-odds of voting for conservatives party.*β*_{2}represents the average difference in log odds of voting for conservatives party between male and female(baseline),holding other constant.*β*_{3}represents the average difference in log odds of voting for conservatives party between people born outside Canada and those who born in Canada(baseline) and ,holding other constant*β*_{4}represents the average difference in log odds of voting for conservatives party between people don’t know if they born in Canada or not and those who born in Canada(baseline) and ,holding other constant*β*_{5}represents the the average difference in log odds of voting for conservatives party between income =125,000 and more and income= 100,000 to 124,999(baseline), holding other constant.*β*_{6}represents the the average difference in log odds of voting for conservatives party between income =25,000 to 49,999 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{7}represents the the average difference in log odds of voting for conservatives party between income =50,000 to 74,999 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{8}represents the the average difference in log odds of voting for conservatives party between income =75,000 to 99,999 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{9}represents the the average difference in log odds of voting for conservatives party between income =Less than 25,000 and income= 100,000 to 124,999(baseline), holding other constant.*β*_{10}represents the the average difference in log odds of voting for conservatives party between people live in British Columbia and people live in Alberta(baseline), holding other constant.*β*_{11}represents the the average difference in log odds of voting for conservatives party between people live in Manitoba and people live in Alberta(baseline), holding other constant.*β*_{12}represents the the average difference in log odds of voting for conservatives party between people live in New Brunswick and people live in Alberta(baseline), holding other constant.*β*_{13}represents the the average difference in log odds of voting for conservatives party between people live in Newfoundland and Labrador and people live in Alberta(baseline), holding other constant.*β*_{14}represents the the average difference in log odds of voting for conservatives party between people live in Nova Scotia and people live in Alberta(baseline), holding other constant.*β*_{15}represents the the average difference in log odds of voting for conservatives party between people live in Ontario and people live in Alberta(baseline), holding other constant.*β*_{16}represents the the average difference in log odds of voting for conservatives party between people live in Prince Edward Island and people live in Alberta(baseline), holding other constant.*β*_{17}represents the the average difference in log odds of voting for conservatives party between people live in Quebec and people live in Alberta(baseline), holding other constant.*β*_{18}represents the the average difference in log odds of voting for conservatives party between people live in Saskatchewan and people live in Alberta(baseline), holding other constant.*β*_{19}represents the the average difference in log odds of voting for conservatives party between people with non-university certificate and people with bachelor’s degree(baseline), holding other constant*β*_{20}represents the the average difference in log odds of voting for conservatives party between people with High school diploma and people with bachelor’s degree(baseline), holding other constant*β*_{21}represents the the average difference in log odds of voting for conservatives party between people with non-university certificate and people with bachelor’s degree(baseline), holding other constant*β*_{22}represents the the average difference in log odds of voting for conservatives party between people with below the bachelor’s and people with bachelor’s degree(baseline), holding other constant*β*_{23}represents the the average difference in log odds of voting for conservatives party between people with degree above the bachelor’s and people with bachelor’s degree(baseline), holding other constant

```
# Conservatives
full_model2 <- glm(vote_cons~age+sex+place_birth_canada+province+hh_size+
income_family+education, data= survey_data,
family="binomial")
summary(full_model2)
# Stepwise Selection based on AIC
sel.var.aic2 <- step(full_model2, trace = 0, k = 2, direction = "both")
sel.var.aic2<-attr(terms(sel.var.aic2), "term.labels")
# selected model
model_cons <- glm(vote_cons~age + sex +place_birth_canada +
income_family+province+education,
data= survey_data,family="binomial")
summary(model_cons)
vif(model_liberal)
vif(model_cons)
```

#### Model Assumption Check

After performing these two logistic models, we need to check four assumptions which are “outcome is binary”, “linearity in the logit for continuous variables”, “absence of multicollinearity” and “lack of strongly influential outliers” separately.

The outcome means results, which is whether respondents vote for Liberal Party or Conservative Party. The outcome has only two value”1” or “0”.

In order to check the assumption ”linearity in the logit for continuous variables”, we need to use Box-Tidwell to check whether there is linearity between logit and age.

Firstly, we need to set the null hypothesis and alternative hypothesis. Null hypothesis means “there is linearity between logit and variable age”. While the alternative hypothesis is the opposite side of the null hypothesis, which there is no linearity between logit and variable age. When we reject the null hypothesis, we conclude this alternative hypothesis.

Lastly, we need to set a 5% cutoff given the null hypothesis is true and we call it level of significance in statistics. The region is named rejection region and it represents the extreme value. If the value falls in the region, we say we reject the null hypothesis, otherwise we fail to reject the null hypothesis.

*H*_{0}: linearity between continuous variable(age) and log-odds
*H*_{A}: non-linearity between continuous variable(age) and
log-odds

```
#linearity
boxTidwell(vote_liberal ~ age, data=survey_data)
```

```
## MLE of lambda Score Statistic (t) Pr(>|t|)
## 3.9192 1.3566 0.175
##
## iterations = 5
```

p-value[23] is the probability of observing extreme value given the null hypothesis is true. After performing the Box-Tidewell, we get p-value 0.175, which is fairly large, meaning that the linearity assumption seems to be satisfied.

We used variance inflation factor to check whether there is a relationship between predictors, which will affect the final results. We assessed these two models. And the values are all approximately 1, which indicates the absence of multicollinearity.

As can seen from the plots above, there was no influential points. This is the final model with five predictors after the variable selection,including numerical variable age, four categorical variables birth place in Canada, province,education degree, and sex.However, for better comparison and anticipation between conservatives party and Liberal party, we decided to add the income family in the final model. Here is the final logistic model with five predictorsweam using to predict the proportion of voters who will vote for the liberal party

#### Limitation of method:

The stepwise selection comes at a heavy price although it is quite helpful to select models. The estimated coefficients from a post-model selection model will be actually biased estimators, and the selected model heavily depends on the data. The method usually result larger test statistic than it should be, leading to reject more than we should(Type I error). Therefore, we need to validate the model to further determine if the model is reasonable for prediction.

## Post-Stratification

After fitting the model, we performed the post-stratification[9] to
get the final results. Post-stratification analysis can provide us with
representative results and improve the accuracy of the survey estimate.
Post-stratification is a statistical method that estimates the value
(possibility) for each bin. **Note that, the word “bin” in statistics
refers to “category” and “cell”.** Based on the research variables, we
divided the entire population into different cells to conduct the
analysis. Then, we used demographics to “extrapolate” how the entire
population will vote.

Here is the formula of the post-stratification:

$$\hat{y}^{P S}=\frac{\sum N_{j} \widehat{y}_{j}}{\sum N_{j}}$$

The formula demonstrates that we multiply the estimate for each bin by its proportion to the entire population. Then we add them together to obtain the final answer.

In our investigation,we divided the census data into different bins based on the seven predictor variables in our model(age, sex, birthplace, province, household size, household income, and education). We applied the model to each bin to predict the estimate. After completing this, we were able to find the proportion of voters from each party at each bin. We then multiplied the proportion with the estimate of each bin. We were able to complete our post stratification analysis by adding the number at each bin together.

## Results

Here is the value of each parameter that we use in the model.

```
as_flextable(model_liberal) %>% autofit() %>% fit_to_width(7.5)
```

Estimate | Standard Error | z value | Pr(>|z|) | ||
---|---|---|---|---|---|

(Intercept) | -2.000 | 0.312 | -6.406 | 0.0000 | *** |

age | 0.010 | 0.003 | 3.177 | 0.0015 | ** |

sexMale | -0.156 | 0.097 | -1.613 | 0.1068 | |

place_birth_canadaBorn outside Canada | 0.657 | 0.129 | 5.090 | 0.0000 | *** |

place_birth_canadaDon't know | 12.692 | 324.744 | 0.039 | 0.9688 | |

provinceBritish Columbia | 0.655 | 0.258 | 2.540 | 0.0111 | * |

provinceManitoba | 0.821 | 0.295 | 2.788 | 0.0053 | ** |

provinceNew Brunswick | 1.252 | 0.316 | 3.962 | 0.0001 | *** |

provinceNewfoundland and Labrador | 1.519 | 0.312 | 4.868 | 0.0000 | *** |

provinceNova Scotia | 1.346 | 0.306 | 4.405 | 0.0000 | *** |

provinceOntario | 1.381 | 0.252 | 5.474 | 0.0000 | *** |

provincePrince Edward Island | 1.450 | 0.308 | 4.710 | 0.0000 | *** |

provinceQuebec | 1.193 | 0.257 | 4.647 | 0.0000 | *** |

provinceSaskatchewan | 0.023 | 0.330 | 0.068 | 0.9455 | |

educationCollege, CEGEP or other non-university certificate or di... | -0.545 | 0.130 | -4.181 | 0.0000 | *** |

educationHigh school diploma or a high school equivalency certificate | -0.307 | 0.162 | -1.898 | 0.0576 | . |

educationLess than high school diploma or its equivalent | -0.509 | 0.245 | -2.074 | 0.0381 | * |

educationUniversity certificate or diploma below the bachelor's level | -0.174 | 0.183 | -0.950 | 0.3422 | |

educationUniversity certificate, diploma or degree above the bach... | 0.111 | 0.138 | 0.804 | 0.4214 | |

income_family$125,000 and more | 0.010 | 0.161 | 0.065 | 0.9482 | |

income_family$25,000 to $49,999 | -0.044 | 0.188 | -0.236 | 0.8134 | |

income_family$50,000 to $74,999 | -0.048 | 0.177 | -0.270 | 0.7872 | |

income_family$75,000 to $99,999 | 0.085 | 0.185 | 0.459 | 0.6462 | |

income_familyLess than $25,000 | -0.233 | 0.202 | -1.152 | 0.2493 | |

Signif. codes: 0 <= '***' < 0.001 < '**' < 0.01 < '*' < 0.05 | |||||

(Dispersion parameter for binomial family taken to be 1) | |||||

Null deviance: 2808 on 2212 degrees of freedom | |||||

Residual deviance: 2625 on 2189 degrees of freedom |

```
as_flextable(model_cons) %>% autofit() %>% fit_to_width(7.5)
```

Estimate | Standard Error | z value | Pr(>|z|) | ||
---|---|---|---|---|---|

(Intercept) | -0.244 | 0.282 | -0.866 | 0.3867 | |

age | 0.012 | 0.003 | 3.738 | 0.0002 | *** |

sexMale | 0.632 | 0.103 | 6.110 | 0.0000 | *** |

place_birth_canadaBorn outside Canada | -0.190 | 0.146 | -1.297 | 0.1946 | |

place_birth_canadaDon't know | -10.938 | 324.744 | -0.034 | 0.9731 | |

income_family$125,000 and more | 0.366 | 0.164 | 2.226 | 0.0260 | * |

income_family$25,000 to $49,999 | -0.248 | 0.195 | -1.277 | 0.2015 | |

income_family$50,000 to $74,999 | -0.151 | 0.182 | -0.832 | 0.4052 | |

income_family$75,000 to $99,999 | -0.132 | 0.195 | -0.679 | 0.4974 | |

income_familyLess than $25,000 | -0.107 | 0.208 | -0.515 | 0.6067 | |

provinceBritish Columbia | -1.922 | 0.216 | -8.895 | 0.0000 | *** |

provinceManitoba | -1.004 | 0.250 | -4.015 | 0.0001 | *** |

provinceNew Brunswick | -1.389 | 0.286 | -4.862 | 0.0000 | *** |

provinceNewfoundland and Labrador | -1.974 | 0.294 | -6.716 | 0.0000 | *** |

provinceNova Scotia | -1.759 | 0.282 | -6.237 | 0.0000 | *** |

provinceOntario | -1.715 | 0.214 | -8.021 | 0.0000 | *** |

provincePrince Edward Island | -1.784 | 0.287 | -6.224 | 0.0000 | *** |

provinceQuebec | -2.485 | 0.230 | -10.784 | 0.0000 | *** |

provinceSaskatchewan | -0.364 | 0.255 | -1.430 | 0.1528 | |

educationCollege, CEGEP or other non-university certificate or di... | 0.558 | 0.132 | 4.212 | 0.0000 | *** |

educationHigh school diploma or a high school equivalency certificate | 0.497 | 0.161 | 3.080 | 0.0021 | ** |

educationLess than high school diploma or its equivalent | 0.807 | 0.234 | 3.453 | 0.0006 | *** |

educationUniversity certificate or diploma below the bachelor's level | 0.237 | 0.188 | 1.264 | 0.2062 | |

educationUniversity certificate, diploma or degree above the bach... | -0.487 | 0.164 | -2.977 | 0.0029 | ** |

Signif. codes: 0 <= '***' < 0.001 < '**' < 0.01 < '*' < 0.05 | |||||

(Dispersion parameter for binomial family taken to be 1) | |||||

Null deviance: 2857 on 2212 degrees of freedom | |||||

Residual deviance: 2488 on 2189 degrees of freedom |

When we ran our model on each of the political parties that we were investigating by post-stratification method, these were the results we got:

```
# Here we will perform the post-stratification calculation
#new data with different bins in census data(I put people in different bins based on predictors(age, sex, place_birth_canada, province, education)),
#Then,we calculate proportion of each cell relative to the whole population(n()/18730)
#Lastly, we use estimate to multiply by the proportion and sum them to get the final answer
#liberal
s_data <- census_data %>% group_by(age, sex, place_birth_canada, province, education, hh_size,
income_family) %>%
summarise(num_people = n(), cell_prop = n()/nrow(census_data))
s_data$estimate_lib <- model_liberal %>% predict(newdata = s_data, type= "response")
s_data <- s_data %>% mutate(lib_pre_prop = estimate_lib * cell_prop)
r1<-sum(s_data$lib_pre_prop)
#conservative
s_data$estimate_cons <- model_cons %>% predict(newdata = s_data, type= "response")
s_data <- s_data %>% mutate(cons_pre_prop = estimate_cons * cell_prop)
r2<-sum(s_data$cons_pre_prop)
```

```
results <- data.frame(Political_Party = c("Liberal Party", "Conservative Party"),
Probability_Winning_Election = c(r1, r2))
names(results)[1] <- "Polical Party"
names(results)[2] <- "Probability of Winning Election"
knitr::kable(results, caption = "Probability of each party winning the election")
```

Polical Party | Probability of Winning Election |
---|---|

Liberal Party | 0.3468515 |

Conservative Party | 0.3563814 |

Probability of each party winning the election

Based on our logistic model, we were able to get the probability of each political party winning the election in 2025. The logistic model indicated that there was a 34.7% probability of the Liberal party winning the election and a 35.6% probability of the Conservative party winning the election. Based on our findings, our data indicates that the Conservative party will win the election in 2025.

However, we must acknowledge that the values for probabilities are only estimates. This is because our model does not take into account the people who haven’t voted. There is still a possibility for the Liberal party to win the election if more people who haven’t voted, end up voting for the Liberal party. Moreover, we used our model only on two of the leading political parties and did not take into account any other political parties. Also, we didn’t have control on the data collection. Thus, there might be some hidden groups that we didn’t take into account.

## Conclusions

Using a logistic model and post-stratification analysis, we found that
the **Conservative Party has a 35.6%** probability of winning the
election in 2025, and the **Liberal Party has a 34.7%** probability of
winning the election. This indicates that based on our model and
analysis, the Conservative Party will win the election in 2025.

However, we need to address that our analysis has certain limitations that we need to take into account. Firstly, we don’t have any control on the survey data ‘2019 phone survey’. This means that we aren’t aware of how the data was collected and how reliable the data is. Moreover, there is always a possibility of a hidden group existing in data collection that we may have not come across. We also did not account for the fact that certain individuals would have chosen not to answer the survey or didn’t decide which party they wanted to vote for. Lastly, our model and analysis was based on only the Liberal party and the Conservative Party. It does not take into account any other political parties in the election.

Therefore, while our model predicted that the Conservative Party will win, there is still a possibility of the Liberal Party or some other political party winning if we take into account the limitations addressed above.

In future analyses, we would suggest using other models to ensure reliability in our analysis and also make sure that we are taking more factors into account when predicting the election outcome in 2025.

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