Week 2: Chi-square Tests for Goodness-of-fit and Independence

Zoom Office Hours: Tuesday/Wednesday/Thursday 1:30 PM - 3:00 PM

1. Chi-square Distribution and Goodness-of-fit Test

Lecture Note: |HTML| PDF|

• Chi-square Distribution
• Chi-square distribution density curve: skewed to the right
• Finding right-tail probability: related to p-value of chi-square tests
• Find percentile: related to critical value.
• Two R built-in functions
• right-tail probability: pchisq(x, df, lower.tail = FALSE)
• percentile: qchisq(p, df, lower.tail = TRUE)
• Chi-square Goodness-of-fit Test
• Setting up Hypothesis
• H₀: The data follows the given distribution p = (p₁, p₂, ..., p_k)
• H_a: The data NOT follow the given distribution p = (p₁, p₂, ..., p_k)
• Calculate expected frequencies under H₀
• Find the total number of observations (i.e.,sample size) denoted by n
• Expected observation of j-th cell: E_j = n* p_j
• Test statistics: $G^2= \sum_{j=1}^k (O_j - E_j)^2/E_j \rightarrow \chi^2_{k-1}$
• Implementation in R: chisq.test(obs.freq, p)
• Observed Frequency: obs.freq = $(n_1, n_2, \cdots, n_k)$
• Hypothetical probability distribution: $p=(p_1, p_2, \cdots, p_k)$

2. Chi-square Test of Independence

Lecture Note: |HTML| PDF|

• Study Designs
• Retropsective corhort study design (look at histoorical data)
• Prospective corhort study design (follow-up study)
• Cross-sectional study (look at the current data, a snapshot observations)
• Measures of Association
• Absolute risk, relative risk, and attributable risk
• Odds ratio
• Risk measures vs study designs
• $\chi^2$ test of independence between two categorical variables
• Observed two-way contingency tables: I rows and J columns
• Null hypothesis: $H_0$ - two categorical variables are independent.
• Expected frequencies under $H_0$:
• Expected frequency of i-th row and j-th column: $E_{ij}=\text{i-th row total}\times \text{j-th column total}/\text{grand total}$
• Test statistic: $G^2 = \sum_{i=1}^I\sum_{j=1}^J (O_{ij}-E_{ij})^2/E_{ij} \rightarrow chi^2_{(I-1)(J-1)}$
• Implementation in R: chisq.test()
• Using observed contingency table: chisq.test(obs.table)
• Using two categorical variables directly: chisq.test(x,y)

3. Weekly Exam #2 Information

• Open: Friday at noon
• Close: Sunday at midnight
• Guideline: |PDF|

Week 1: Computing Software and Review of Introductory Statistics

Zoom Office Hours: Tuesday/Wednesday/Thursday 1:30 PM - 3:00 PM

1. Course Information and Learning Advice

Note: |HTML| PDF|

2. Getting started with R and RStudio

Lecture Note: |HTML| PDF|

• Install the current version of R. It is freely available at https://www.r-project.org/
• Install the free version of RStudio. The downloading site is at https://posit.co/products/open-source/rstudio/?sid=1
• Use R as a super graphing calculator
• Bult-in R functions for descriptive statistics
• Base R packages
• Writing simple and re-usable R scripts

3. Basic statistics review

Lecture Note: |HTML| PDF|

• Sampling distributions and Central Limit Theorem (CLT)
• Confidence intervals of sample means: normal and t distributions
• Testing hypothesis: logic, steps, p-value

4. Least square simple linear regression (SLR): inference and applications

Lecture Note: |HTML| PDF|

• Structure, interpretation of coeffcients with an emphasis on the slope parameters
• Assumptions and model diagnostics (validation)
• R functions for creating various residual diagnostic plots
• Clear understanding of R output and know hoe to use the information to
• perform hypothesis testing on the slope
• construct confidence interval of the slope
• assess the goodness-of-fit using R²

5. Weekly Exam #1

• Information: |PDF|
• Answer Key and Summary of Exam #1: |PDF|