Problem Definition

Significance of Research

• The findings from the research will provide critical information that the management can use to increase loan/ratio rate and increase profit margin.
• Thus, the the research is critical in turning around the company’s struggling financial status.

Information gathering

Reasons for Consumers’ Use of Credit Union

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “research question” because you do not have enough evidence to state the relationship with confidence.

R1: Why do people join the Credit Union?

Using services from Other Financial Services

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “research question” because you do not have enough evidence to state the relationship with confidence.

R2: Why do members use other financial institutions when they need to borrow funds?

Attitudes and beliefs about the employee proficiency

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “research question” because you do not have enough evidence to state the relationship with confidence.

R3: What are the members’ attitudes and beliefs about the proficiencies of credit union employees?

Differences in Headquarters vs. Elswhere

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “research question” because you do not have enough evidence to state the relationship with confidence.

R4: Are there any perceived differences on attitude towards employee proficiencies, awareness of services, and operational effectiveness between members who live in the area of the firm’s headquarters and members who live elsewhere?

Awareness of Services

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “research question” because you do not have enough evidence to state the relationship with confidence.

R5: How well are the members aware of the services offered by the credit union?

CU’s Operational Effectiveness

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “research question” because you do not have enough evidence to state the relationship with confidence.

R6: What are the members’ attitudes and beliefs about how effectively the credit union is operated?

Adequancy of Finacial Services

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “Hypothesis” when you have enough evidence to state the relationship with confidence.

H1: Members’ awareness of the regular share accounts will influence their opinions on the adequacy of financial services in meeting members’ needs.

CU Members’ Opinion on CU’s Loan Rate

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “Hypothesis” when you have enough evidence to state the relationship with confidence.

H2: Members will not be different in their opinion on CU’s loan rate (Q8r) and the degree to which they agree that CU’s loan rate is lower than competitors (Q30).

Impact of Current loan use and residence lcoatoin on Finacial Services Adequacy

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “Hypothesis” when you have enough evidence to state the relationship with confidence.

Loan Status and CU Loan Rate

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “Hypothesis” when you have enough evidence to state the relationship with confidence.

Factors Affecting Attitudes towards Credit Union Operational Efficiency

• What to write on this sub-section?
  • You can use information you learned from academic research
  • You may use white paper available online
  • You can use information you obtained from managers
• Weave the information from various sources to form a logic to support the relationships among the variables in the research question.
• You are labeling it “Hypothesis” when you have enough evidence to state the relationship with confidence.

H5a: Member belief about CU keeps personal financial information confidential will positively influence the overall attitude toward credit union’s operational efficiency.

Research Design

• A mail survey was used due to the following reasons:
  • CU membership is widely dispersed geographically
  • Low cost
  • Sensitive questions needs anonymity to induce a candid response
  • Long time needed for mail survey is not an issue for the board.

Population and Sampling

• Population: All current members of the EFCU.
• Sampling frame:
  • A list of members and their addresses is available.
  • 3,531 members as of January 31 for trial balance listing of the EFCU membership

Data Collection and coding

Data Preparation

Variables in the data

• Number of variables: 54

Sample Characteristics

• Number of participants: 128
• % of the participants who live in the headquarter area vs. non-headquarter area: 44.5% vs. 55.5%.
• No demographic variables are available on the survey
  • Gender
  • Ethnicity/Race
  • Income
  • Education

Measures

• Refer to the Construct and Measurement Table.
• All constructs, operationalization, and scales will be provided later.

Descriptive Statistics

Figure 3: Histogram for Savings Rate (Q7r)

One-sample t-test

mean_q7r <- case31 |> #count(q7r)
  summarise(mean = mean(q7r, na.rm = TRUE)) |> 
  round(2)

case31 |> 
  t.test(q7r ~ 1, mu = 3, data = _)

    One Sample t-test

data:  q7r
t = 1.3016, df = 117, p-value = 0.1956
alternative hypothesis: true mean is not equal to 3
95 percent confidence interval:
 2.964639 3.170954
sample estimates:
mean of x 
 3.067797 

Descriptive Statistics

One-sample t-test

Descriptive Statistics

Means & SD

describe(case31$at_emp)

   vars   n mean   sd median trimmed  mad  min max range  skew kurtosis   se
X1    1 128 4.21 0.59      4    4.25 0.74 2.75   5  2.25 -0.32    -0.69 0.05

Histogram

Figure 4: Histogram of Employee Proficiencies (at_emp) with a Mean line

Reliability/factor analysis

• We already looked at reliability and unidimensionality in the data preparatoin stage.
• The four items are highly representative of the latent construct.
• Therefore, we created a composite index and use it

One-sample t-test

case31 |> 
  t.test(at_emp ~ 1, mu = 4, data = _)

    One Sample t-test

data:  at_emp
t = 3.9063, df = 127, p-value = 0.0001515
alternative hypothesis: true mean is not equal to 4
95 percent confidence interval:
 4.101192 4.308965
sample estimates:
mean of x 
 4.205078 

R4a Analysis Overview (Two samples t-test)

R4a: Are there any perceived differences on attitude towards employee proficiency between members who live in the area of the firm’s headquarters and members who live elsewhere?

Descriptives

Boxplot

Figure 5: Attitudes toward employee proficiency by residence

Barplot

Figure 6: Attitudes toward employee proficiency by residence

Two independent samples t-test

library(car)

case31 |> 
  mutate(q38 = haven::as_factor(q38)) |> 
  leveneTest(at_emp ~ q38, data = _)

Levene's Test for Homogeneity of Variance (center = median)
       Df F value Pr(>F)
group   1   0.041 0.8399
      126               

case31 |> 
  mutate(q38 = haven::as_factor(q38)) |> 
  var.test(at_emp ~ q38, data = _)

    F test to compare two variances

data:  at_emp by q38
F = 0.89996, num df = 56, denom df = 70, p-value = 0.6864
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
 0.548864 1.496643
sample estimates:
ratio of variances 
         0.8999608 

# t.test: assuming var are equal
case31 |> 
  t.test(at_emp ~ q38, data = _, var.equal = TRUE) 

    Two Sample t-test

data:  at_emp by q38
t = 1.2941, df = 126, p-value = 0.198
alternative hypothesis: true difference in means between group 1 and group 2 is not equal to 0
95 percent confidence interval:
 -0.0721542  0.3448253
sample estimates:
mean in group 1 mean in group 2 
       4.280702        4.144366 

Summary Table

case31 |> 
  mutate(q38 = haven::as_factor(q38)) |> 
  tbl_summary(
    include = c(q38, at_emp),
    by = q38,
    type = at_emp ~ "continuous2",
    statistic = list(all_continuous() ~ c("{mean} ({sd})",
                                          "{min}, {max}"))
  ) |> 
  add_difference() |> 
  add_overall() |> 
  add_n() |> 
  modify_caption("**Test of Differences in Attitudes toward Employee Proficiency between customers who live in Headquarter area and those who live in non-headquarter area**") |> 
  modify_spanning_header(
    c(stat_1, stat_2) ~ "**Live in Headquarter Area?**") 

R4b Analysis Overview (\(\chi^2\) Independence Test)

R4b: Are there any perceived differences on awareness of services between members who live in the area of the firm’s headquarters and members who live elsewhere?

q13r (Share account) by q38

• Cleaning
• cross-tab
• Mosaic Plot
• Chi-square test of independence

case31 <- case31 |> 
  mutate(q13r = haven::as_factor(q13r),
         q38 = haven::as_factor(q38))

table(case31$q13r, case31$q38)

                         
                          yes no
  Unaware                  34 49
  Aware but Have Not Used  16 16
  Aware and Have Used       7  6

# cross-tab: %
table(case31$q13r, case31$q38) |> 
  prop.table(margin = 2) * 100 

                         
                                yes        no
  Unaware                 59.649123 69.014085
  Aware but Have Not Used 28.070175 22.535211
  Aware and Have Used     12.280702  8.450704

• \(H_0\): Awareness of regular share account is independent of members’ residence
• \(H_A\): Awareness of regular share account is dependent on members’ residence

table(case31$q13r, case31$q38) |>
  chisq.test() 

    Pearson's Chi-squared test

data:  table(case31$q13r, case31$q38)
X-squared = 1.2717, df = 2, p-value = 0.5295

q14r (Special subaccounts) by q38

case31 <- case31 |> 
  mutate(q14r = haven::as_factor(q14r))

# cross-tabulation: count

table(case31$q14r, case31$q38)

                         
                          yes no
  Unaware                  22 24
  Aware but Have Not Used  31 39
  Aware and Have Used       4  8

# cross-tab: %
table(case31$q14r, case31$q38) |> 
  prop.table(margin = 2) * 100 

                         
                                yes        no
  Unaware                 38.596491 33.802817
  Aware but Have Not Used 54.385965 54.929577
  Aware and Have Used      7.017544 11.267606

table(case31$q14r, case31$q38) |>
  chisq.test() 

    Pearson's Chi-squared test

data:  table(case31$q14r, case31$q38)
X-squared = 0.81305, df = 2, p-value = 0.666

q15r by q38 with gt_summary table

• q15r (Christmas club account)

case31 <- case31 |> 
  mutate(q15r = haven::as_factor(q15r))

case31 |> 
  tbl_summary(
    include = c(q15r, q38),
    by = q38
  ) |> 
  add_p() |> 
  modify_spanning_header(all_stat_cols() ~ "**Live in HQ Area?**") |> 
  add_n() |> 
  italicize_labels() |> 
  #show_header_names()
  modify_header(label ~ "**Awareness**") |> 
  as_gt() |> 
  tab_header(
    title = md("**Chi-Square Test of Independece**"),
    subtitle = md("*Christmas Club Account Awareness by Where Customers Live*")
  )

Further cleaning

case31_clean <- case31 |> 
  mutate(q13r = haven::as_factor(q13r),
         q14r = haven::as_factor(q14r),
         q15r = haven::as_factor(q15r),
         q16r = haven::as_factor(q16r),
         q17r = haven::as_factor(q17r),
         q18r = haven::as_factor(q18r),
         q19r = haven::as_factor(q19r),
         q20r = haven::as_factor(q20r),
         q21r = haven::as_factor(q21r),
         q22r = haven::as_factor(q22r),
         q23r = haven::as_factor(q23r),
         q24r = haven::as_factor(q24r),
         q25r = haven::as_factor(q25r),
         q26r = haven::as_factor(q26r)
         )

Creating a function for chi-square independence test

chisq_ind_test <- function(data, var) {
  
  library(dplyr)
  library(rlang)
  library(glue)
  
  var <- enquo(var)  # Capture the column as a quosure
  
  # Remove NA's
  data <- data |> 
    filter(!is.na(!!var), !is.na(q38))
  
  # Remove unused levels to avoid errors in chi-square test
  data <- data |> 
  mutate(!!var := droplevels(!!var)) # dynamic column name
  
  # Cross-tab
  tab <- table(pull(data, !!var), data$q38)

  # Cross-tab: %
  prop_tab <- prop.table(tab, margin = 2) * 100

  # Mosaic Plot
  tab_c <- table(data$q38, pull(data, !!var))
  
  mosaicplot(
    tab_c,
    main = glue::glue("Awareness of {quo_name(var)} by Residence"),
    color = TRUE, las = 1
  )

  # Chi-square test of independence
  chi_sq_result <- chisq.test(tab)

  # Return results as a list
  list(
    crosstab = tab,
    proportions = prop_tab,
    chi_square = chi_sq_result
  )
}

Testing all

awareness <- vars(q13r, q14r, q15r, q16r, q17r, q18r, q19r, q20r, q21r, q22r, q23r, q24r, q25r, q26r) # Quote the variables

awareness |> 
  map(~ chisq_ind_test(case31_clean, !!.x))

testing for q26r by q38

case31_clean |> 
  chisq_ind_test(q26r)

$crosstab
                         
                          yes no
  Aware but Have Not Used  37 43
  Aware and Have Used      20 28

$proportions
                         
                               yes       no
  Aware but Have Not Used 64.91228 60.56338
  Aware and Have Used     35.08772 39.43662

$chi_square

    Pearson's Chi-squared test with Yates' continuity correction

data:  tab
X-squared = 0.10332, df = 1, p-value = 0.7479

R4c Analysis Overview

• R4c: Are there any perceived differences on operational effectiveness between members who live in the area of the firm’s headquarters and members who live elsewhere?

q13r

Summary table

case31_clean |> 
  tbl_summary(
    include = q13r,
    type = all_categorical() ~ "categorical"
  ) |> 
  italicize_labels() |> 
  #show_header_names()
  modify_header(label ~ "**Awareness**") |> 
  modify_caption("**Chi-Square Goodness of Fit Test**:<br>*Regular Share Account Awareness*")

Barplot

case31_clean |>
  ggplot(aes(q13r, fill = q13r)) +
  geom_bar() +
  theme(legend.position = "none")

Chi-square Goodness-of-fit test

• \(H_0\) = the observed data follows the expected distribution
• \(H_A\) = the observed data follows the expected distribution

expected = c(1/3, 1/3, 1/3)
table(case31_clean$q13r) |> 
  chisq.test(p = expected)

    Chi-squared test for given probabilities

data:  table(case31_clean$q13r)
X-squared = 61.422, df = 2, p-value = 4.596e-14

More efficient testing with a function

# data

awareness_long <- case31_clean |> 
  select(q13r:q26r) |> 
  pivot_longer(cols = everything(),
               names_to = "services",
               values_to = "aware_level")
awareness_long

# A tibble: 1,792 × 2
   services aware_level            
   <chr>    <fct>                  
 1 q13r     Aware and Have Used    
 2 q14r     Aware but Have Not Used
 3 q15r     Aware but Have Not Used
 4 q16r     Aware but Have Not Used
 5 q17r     Aware but Have Not Used
 6 q18r     Aware but Have Not Used
 7 q19r     Aware and Have Used    
 8 q20r     Aware and Have Used    
 9 q21r     Aware but Have Not Used
10 q22r     Aware but Have Not Used
# ℹ 1,782 more rows

Visualization

Frequency by groups

Figure 8: Frequency Distribution of Adequacy of Financial Services in Meeting Members’ Needs (q33) by Their Awareness of Regular Share Accounts (q13r)

Mean value by groups

Figure 9: Mean VAlue of Adequacy of Financial Services in Meeting Members’ Needs (q33) by Their Awareness of Regular Share Accounts (q13r)

ANOVA Test Procedure

1. Check normality assumption

• The dependent variable should be approximately normally distributed within each group.
• \(H_0:\) The data are normally distributed
• \(H_1:\) The data are NOT normally distributed

# Shapiro-Wilk normality test 

case31_clean |> 
  pull(q33) |> 
  shapiro.test()

    Shapiro-Wilk normality test

data:  pull(case31_clean, q33)
W = 0.80776, p-value = 1.176e-11

case31_clean |> 
  group_by(q13r) |> 
  summarise(p_value = shapiro.test(pull(cur_data(), q33))$p.value)

# A tibble: 3 × 2
  q13r                          p_value
  <fct>                           <dbl>
1 Unaware                 0.00000000133
2 Aware but Have Not Used 0.0000121    
3 Aware and Have Used     0.0136       

h1_aov_model <- case31_clean |> 
  aov(q33 ~ q13r, data = _) 

residuals(h1_aov_model) |> 
  shapiro.test()

    Shapiro-Wilk normality test

data:  residuals(h1_aov_model)
W = 0.87843, p-value = 7.927e-09

2. Homogeneity of variance

• \(H_0:\) Variance is the same across the groups
• \(H_1:\) The data is NOT normally distributed

library(car)
case31_clean |> 
  leveneTest(q33 ~ q15r, data = _)

Levene's Test for Homogeneity of Variance (center = median)
       Df F value Pr(>F)
group   2  1.0002 0.3707
      125               

3. ANOVA Test

• \(H_0:\) All group means are equal (\(\mu_1 = \mu_2 = \mu_3 = ... = \mu_k\))
• \(H_1:\) At least one pair of group means is different

case31_clean |> 
  aov(q33 ~ q13r, data = _) |> 
  summary()

             Df Sum Sq Mean Sq F value Pr(>F)  
q13r          2   4.06  2.0307   4.631 0.0115 *
Residuals   125  54.81  0.4385                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

case31_clean |> 
  kruskal.test(q33 ~ q13r, data = _)

    Kruskal-Wallis rank sum test

data:  q33 by q13r
Kruskal-Wallis chi-squared = 7.1098, df = 2, p-value = 0.02858

4. Summary Table

anova_model <- case31_clean |> 
  aov(q33 ~ q13r, data = _) 

anova_summary <- summary(anova_model)

anova_pvalue <- anova_summary[[1]][["Pr(>F)"]][1]

case31_clean |> 
  mutate(q33 = as.numeric(q33)) |> 
  tbl_summary(
    include = c(q13r, q33),
    by = q13r,
    type = q33 ~ "continuous2",
    statistic = list(all_continuous() ~ c("{mean} ({sd})",
                                          "{min}, {max}")),
    label = q33 ~ "Adequacy of Financial Services"
  ) |> 
  modify_spanning_header(
    all_stat_cols() ~ "**Regular Share Account Awareness Levels**") |> 
  add_overall() |> 
  modify_caption("**Test of Differences in Perceived Adequacy of Financial Services in Meeting Memberss Needs:**<br>_Diffential Effect by Awareness of Regular Share Account (q13r)_") |> 
  italicize_labels() |> 
  modify_footnote(
    c(stat_1, stat_2, stat_3) ~ glue::glue("One-way ANOVA with equal variance assummed p-value: {format.pval(anova_pvalue, digits = 3)}")) |> 
  #show_header_names()
  modify_header(label ~ "**Variable**")

5. Post-hoc Analysis if not all group means are the same

case31_clean |> 
  aov(q33 ~ q13r, data = _) |> 
  TukeyHSD()

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = q33 ~ q13r, data = case31_clean)

$q13r
                                                   diff        lwr         upr
Aware but Have Not Used-Unaware              0.09224398 -0.2345927  0.41908065
Aware and Have Used-Unaware                 -0.54958295 -1.0180950 -0.08107087
Aware and Have Used-Aware but Have Not Used -0.64182692 -1.1584282 -0.12522563
                                                p adj
Aware but Have Not Used-Unaware             0.7816413
Aware and Have Used-Unaware                 0.0170339
Aware and Have Used-Aware but Have Not Used 0.0106252

library(dunn.test)

# Post-hoc pairwise comparisons
dunn.test(as.numeric(case31_clean$q33), case31_clean$q13r, method = "bonferroni")

  Kruskal-Wallis rank sum test

data: x and group
Kruskal-Wallis chi-squared = 7.1098, df = 2, p-value = 0.03

                           Comparison of x by group                            
                                 (Bonferroni)                                  
Col Mean-|
Row Mean |   Aware an   Aware bu
---------+----------------------
Aware bu |  -2.534359
         |    0.0169*
         |
 Unaware |  -2.501555   0.419916
         |    0.0185*     1.0000

alpha = 0.05
Reject Ho if p <= alpha/2

Checking Assumptions of Multiple Regression

1. Linearity Check

# Scatterplot for each predictor against the outcome
ggplot(case31_clean, aes(x = q31, y = q37r)) + 
  geom_point() + 
  geom_smooth(col = "blue") +
  labs(title = "Linearity Check: q31 vs q37r")

# Scatterplot for each predictor against the outcome
ggplot(case31_clean, aes(x = q32, y = q37r)) + 
  geom_point() + 
  geom_smooth(col = "blue") +
  labs(title = "Linearity Check: q32 vs q37r")

# Scatterplot for each predictor against the outcome
ggplot(case31_clean, aes(x = q33, y = q37r)) + 
  geom_point() + 
  geom_smooth(col = "blue") +
  labs(title = "Linearity Check: q33 vs q37r")

# Scatterplot for each predictor against the outcome
ggplot(case31_clean, aes(x = q34, y = q37r)) + 
  geom_point() + 
  geom_smooth(col = "blue") +
  labs(title = "Linearity Check: q34 vs q37r")

library(car)
lm_model <- case31_clean |> 
  filter(!is.na(q37r)) |> 
  lm(q37r ~ q31 + q32 + q33 + q34, data = _)
crPlots(lm_model)

2. Independence Check

• \(H_0:\) There is no first-order autocorrelation in the residuals (errors are independent).
• \(H_1:\) There is evidence of positive or negative autocorrelation.
• Test statistics (d)
  • \(d \approx 2\): No autocorrelation
  • \(d < 2\): Positive autocorrelation
  • \(d > 2\): Negative autocorrelation

library(lmtest)
dwtest(lm_model)  # Durbin-Watson test for autocorrelation

    Durbin-Watson test

data:  lm_model
DW = 2.2276, p-value = 0.8995
alternative hypothesis: true autocorrelation is greater than 0

3. Homoscedasticity Check

plot(lm_model, which = 1)  # Built-in residuals vs. fitted plot

library(lmtest)
bptest(lm_model)  # Breusch-Pagan test for heteroscedasticity

    studentized Breusch-Pagan test

data:  lm_model
BP = 4.5897, df = 4, p-value = 0.332

4. Normality of Residuals

residuals <- residuals(lm_model)
hist(residuals, breaks = 20, main = "Histogram of Residuals")

qqnorm(residuals)
qqline(residuals, col = "blue")

shapiro.test(residuals)  # Shapiro-Wilk test

    Shapiro-Wilk normality test

data:  residuals
W = 0.95406, p-value = 0.0004082

5. Multicollinearity Check

• VIF > 5: Possible multicollinearity.
• VIF > 10: Strong multicollinearity, consider dropping or combining predictors.

library(car)
vif(lm_model)  # VIF values

     q31      q32      q33      q34 
1.148240 1.396267 1.065648 1.214352 

6. Outliers and Influential Observations

plot(lm_model, which = 4)  # Built-in Cook's distance plot

# Leverage Plot
hatvalues <- hatvalues(lm_model)
plot(hatvalues, main = "Leverage Values", ylab = "Leverage")

# DFBETAS
dfbetas <- dfbeta(lm_model)
head(dfbetas)  # Shows influence of each observation

   (Intercept)           q31           q32          q33          q34
1  0.009403795 -0.0001015945  0.0001734095 -0.001583996 -0.001698192
2  0.018803471 -0.0047139437  0.0022874976 -0.002830564  0.001025494
3 -0.064176774  0.0078605827  0.0010204438  0.004270726  0.002766490
4  0.009403795 -0.0001015945  0.0001734095 -0.001583996 -0.001698192
5  0.049299034 -0.0005326053  0.0009090924 -0.008304040 -0.008902708
6 -0.059692744  0.0126198994 -0.0041741897  0.002834174  0.005853614

7. Cross-validation

library(caret)
set.seed(123)

train_control <- trainControl(method = "cv", number = 10)

case31_clean <- case31_clean |> 
  filter(!is.na(q37r)) |> 
  mutate(q37r = as.numeric(q37r))

cv_model <- train(
  q37r ~ q31 + q32 + q33 + q34,
  data = case31_clean,
  method = "lm",
  trControl = train_control)

cv_model  # Check cross-validation results

Linear Regression 

121 samples
  4 predictor

No pre-processing
Resampling: Cross-Validated (10 fold) 
Summary of sample sizes: 108, 109, 109, 109, 109, 109, ... 
Resampling results:

  RMSE       Rsquared   MAE     
  0.6080421  0.3155103  0.483605

Tuning parameter 'intercept' was held constant at a value of TRUE

cv_model$resample

        RMSE   Rsquared       MAE Resample
1  0.3845295 0.61980439 0.3360649   Fold01
2  1.0946785 0.10531564 0.8239592   Fold02
3  0.4962708 0.03110875 0.4335815   Fold03
4  0.6498566 0.02311003 0.4974842   Fold04
5  0.4871213 0.46218519 0.3913416   Fold05
6  0.6067156 0.26438525 0.4945987   Fold06
7  0.5833334 0.60333089 0.4652271   Fold07
8  0.5420575 0.54103610 0.4160615   Fold08
9  0.4332916 0.44055235 0.3583005   Fold09
10 0.8025663 0.06427416 0.6194310   Fold10

Model Parameters

summary(cv_model) # summary function returns final model fit based on entire data

Call:
lm(formula = .outcome ~ ., data = dat)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.70193 -0.24939  0.00387  0.38004  1.29807 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.34721    0.48011   2.806 0.005884 ** 
q31          0.18835    0.08901   2.116 0.036472 *  
q32          0.30787    0.08302   3.709 0.000321 ***
q33          0.29420    0.08650   3.401 0.000921 ***
q34         -0.06829    0.07917  -0.863 0.390138    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.6186 on 116 degrees of freedom
Multiple R-squared:  0.3006,    Adjusted R-squared:  0.2764 
F-statistic: 12.46 on 4 and 116 DF,  p-value: 1.823e-08

case31_clean |> 
  lm(q37r ~ q31 + q32 + q33 + q34, data = _)  |> 
  summary()

Call:
lm(formula = q37r ~ q31 + q32 + q33 + q34, data = case31_clean)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.70193 -0.24939  0.00387  0.38004  1.29807 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.34721    0.48011   2.806 0.005884 ** 
q31          0.18835    0.08901   2.116 0.036472 *  
q32          0.30787    0.08302   3.709 0.000321 ***
q33          0.29420    0.08650   3.401 0.000921 ***
q34         -0.06829    0.07917  -0.863 0.390138    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.6186 on 116 degrees of freedom
Multiple R-squared:  0.3006,    Adjusted R-squared:  0.2764 
F-statistic: 12.46 on 4 and 116 DF,  p-value: 1.823e-08

Summary Table