Introduction

• Education has always been a personal interest area. This assignment has provided an opportunity to drill into this topic deeper with a statistical lense.
• Upon browsing through www.data.vic.govt.au, I came across the results from the Student Entrant Health Questionnaire for period 2009-2015, which is a survey the parents of all children entering primary school in Victoria need to complete.
• The Questionnaire is primarily intended for School Nurses and covers a range of issues, “including general health, speech, hearing, vision, development, behaviour, well-being and family stress” (Department of Education and Training 2014, para. 2).
• This report will focus only on family stress and speech, and will explore the relationship between the two variables.
• Rpubs link: http://rpubs.com/Raina/429035

Problem Statement

• The purpose of this report is to examine the association between family stress and children’s speech difficulties as reported by the parents. Do children experiencing higher family stress tend to have greater speech difficulties?
• Using data from the Student Entrant Health Questionnaire, a linear regression analysis will be performed to determine if there is a linear relationship between the two variables.
• Family stress is identified as the predictor variable \(x\), and speech difficulties is identified as the dependent variable \(y\).

Data

Data Source

• Open Data is used in this report, which can be easily found on www.data.vic.govt.au.
• Titles of the datasets are: “VCAMS children with high levels of family stress” and “VCAMS children entering school with basic skills for life and learning”.

Sampling Method

• Upon entering primary school in all Local Government Areas (LGA) in Victoria, the parents of all children are requested to fill out the Questionnaire with their observations on a range of issues, which is then used by School Nurses, educators and the Department of Education and Early Childhood Development.

Data Cont.

Variables

• Stress: Proportion of children at school entry whose parents report high levels of family stress in the past month
  • This was a manually added variable by dividing Numerator_Stress (count of children) over Denominator (count of completed surveys)
• Speech: Proportion of children at school entry whose parent report one or more concerns with child speech or language
  • This was a manually added variable by dividing Numerator_Speech (count of children) over Denominator (count of completed surveys)

Preprocessing

• As the Data Published Date, Data Updated Date and Date Temporal Coverage are identical, I believe it is safe to assume the two datasets are actually subsets of the same parent dataset, thus I combined the two datasets into one by matching the Year and LGA_Key.

Descriptive Statistics and Visualisation

• An initial scatter plot was created, but the linearity is not very clear.

SEHQ <- read_csv("/Users/raina/Documents/SEHQ.csv",col_types = cols(Speech = col_number(), 
    Stress = col_number()))

plot(Speech~Stress,data = SEHQ)

Descriptive Statistics and Visualisation Cont.

• I then looked into histograms of the two variables. Turns out both Speech and Stress are somewhat skewed to the right.

par(mfrow=c(1,2))
SEHQ$Speech %>% hist(main="Speech")
SEHQ$Stress %>% hist(main="Stress")

Descriptive Statistics and Visualisation Cont.

• Therefore I decided to transform using sqrt().

par(mfrow=c(1,2))
SEHQ$Speech %>% sqrt() %>% hist(main="sqrt(Speech)")
SEHQ$Stress %>% sqrt() %>% hist(main="sqrt(Stress)")

Descriptive Statistics and Visualisation Cont.

• The new scatter plot looks ever so slightly better.

plot(sqrt(Speech)~sqrt(Stress),data=SEHQ,ylim=c(0.1,0.7),xlim=c(0.1,0.6))

Hypothesis Testing

• Testing for linear regression using the F-test:
  • \(H_0:\) The data does not fit the linear regression model.
  • \(H_A:\) The data fits the linear regression model.
• The p-value for the F-test is very small, F(1,545)=96.31, p<.001
• I can reject \(H_0\); the model is statistically significant.

model <- lm(sqrt(Speech) ~ sqrt(Stress), data = SEHQ)
model %>% summary()

## 
## Call:
## lm(formula = sqrt(Speech) ~ sqrt(Stress), data = SEHQ)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.116567 -0.031320 -0.003927  0.028647  0.142236 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.21524    0.01814  11.863   <2e-16 ***
## sqrt(Stress)  0.52764    0.05377   9.814   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04529 on 545 degrees of freedom
##   (12 observations deleted due to missingness)
## Multiple R-squared:  0.1502, Adjusted R-squared:  0.1486 
## F-statistic: 96.31 on 1 and 545 DF,  p-value: < 2.2e-16

Hypthesis Testing Cont.

• Next, I will test the model parameters using the F-test:
  • Intercept:
    • \(H_0: \alpha = 0\)
    • \(H_A: \alpha \ne 0\)
  • Slope:
    • \(H_0: \beta = 0\)
    • \(H_A: \beta \ne 0\)
• Both p-values are really small, p<.001, and the 95% CIs do not capture \(H_0: \alpha = 0\) and \(H_0: \beta = 0\).
• I can reject both \(H_0\); the model parameters are statistically significant.

model %>% summary()

## 
## Call:
## lm(formula = sqrt(Speech) ~ sqrt(Stress), data = SEHQ)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.116567 -0.031320 -0.003927  0.028647  0.142236 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.21524    0.01814  11.863   <2e-16 ***
## sqrt(Stress)  0.52764    0.05377   9.814   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04529 on 545 degrees of freedom
##   (12 observations deleted due to missingness)
## Multiple R-squared:  0.1502, Adjusted R-squared:  0.1486 
## F-statistic: 96.31 on 1 and 545 DF,  p-value: < 2.2e-16

model %>% confint()

##                  2.5 %    97.5 %
## (Intercept)  0.1796031 0.2508817
## sqrt(Stress) 0.4220293 0.6332534

Hypthesis Testing Cont.

• Same set of assumptions apply to both the linear regression F-test and the model parameters F-tests:
  • Independence
  • Linearity
  • Normality of Residuals
  • Homoscedasticity
• I will validate all of them before reporting the final regression model.

Hypthesis Testing Cont.

• Residuals vs Fitted
  • Trend line is fairly flat in the middle, curvature on both ends could be due to outliers.
  • Variability on y-axis is fairly constant across the range of values on the x-axis.

model %>% plot(which=1)

Hypthesis Testing Cont.

• Normal Q-Q
  • Residuals fall very closely to the line.

model %>% plot(which=2)

Hypthesis Testing Cont.

• Scale-Location
  • The red line is fairly flat.
  • No obvious patterns to the variability of the residuals.

model %>% plot(which=3)

Hypthesis Testing Cont.

• Residuals vs Leverage
  • No values fall close to band .5, in fact, band .5 is not even visible.
  • No influential cases/outliers.

model %>% plot(which=5)

Hypthesis Testing Cont.

• Correlation Coefficient \(r\):
  • \(r\) = 0.3875, indicating a relatively weak positive correlation between the two variables

r <- cor(sqrt(SEHQ$Speech),sqrt(SEHQ$Stress),use = "complete.obs")
r

## [1] 0.3875291

CIr(r=r,n=547,level=.95)

## [1] 0.3138915 0.4565325

Hypthesis Testing - Interpretation

• Simple linear regression summary:
  • Linearity was assumed, normality of residuals ok, homoscedasticity ok, no influential cases.
  • \(r\) = 0.3875, \(r^2\) = 0.1502
  • Model ANOVA, F(1,545)=96.31, p<.001
  • \(\alpha\) = 0.215, p<.001, 95% CI (0.1796,0.2509)
  • \(\beta\) = 0.528, p<.001, 95% CI (0.4220,0.6330)
• Decisions:
  • Overall model: reject \(H_0\)
  • Intercept: reject \(H_0\)
  • Slope: reject \(H_0\)
  • sqrt(Speech) = 0.215 + 0.528 x sqrt(Stress)
• Conclusion:
  • There is a statistically significant positive linear relationship between family stress level and a child’s speech difficulties.

Discussion

• Major finding: By fitting a linear regression model, the investigation demonstrated evidence of a positive linear relationship between family stress level and a child’s speech difficulties. In other words, children experiencing higher family stress level seem to have greater speech difficulties.
• Strengths: Dataset is of a reasonable size, spanning across 6 years and 80 local government areas.
• Limitations: Survey tends to have response bias and/or question design issues.
• Future investigations and beyond: Family stress level would not be the only contributing factor to a child’s speech difficulties. Future analysis could be extended to include other variables, such as genetics, upbringing. More importantly, any investigations would be pointless if no actions were taken by educators and education authorities to try to address the root causes and improve children’s overall development and wellbeing.

References

• Department of Education and Training 2014, VCAMS children with high levels of family stress, datavic, viewed 13 October 2018, https://www.data.vic.gov.au/data/dataset/vcams-children-with-high-levels-of-family-stress.
• Department of Education and Training 2014, VCAMS children entering school with basic skills for life and learning, datavic, viewed 13 October 2018, https://www.data.vic.gov.au/data/dataset/vcams-children-entering-school-with-basic-skills-for-life-and-learning.