OMIS 2050 - Class 8 - Analyzing Univariate Models with Continuous Outcomes

What are the regression model assessment assumptions?

1. model evaluation

• linearity

2. residual analysis

• normality

3. individual test

• homoscedasticity

What is the 1st regression model assessment?

model evaluation

• continue if 1+ significant variable

What is the 2nd regression model assessment?

residual analysis

• diagnose for outlier

What is the 3rd regression model assessment?

variable test

• test significance

What is the sum of squares error formula?

SSE=\sum\left(y_{i}-\hat{y}_{i}\right)^2

What is the sum of squares regression formula?

SSR=\sum\left(\hat{y}_{i}-\bar{y}\right)^2

What is the total sum of squares formula?

SST=\sum\left(y_{i}-\bar{y}\right)^2

What is the total sum of squares formula with the sum of squares error & sum of squares regression?

SST=SSE+SSR

What is the total sum of squares?

|:| SST
total Y variation

What is the sum of squares error?

|:| SSE
unexplained Y variation

What is the sum of squares regression?

|:| SSR

explained Y variation

What is the SSE’s goal?

minimize squared deviation total

What is the SSR & SSE relation?

• X & Y link > → SSR >> SSE

• X & Y ≠ link → SSR 0, SSE 100%

» SSE < → regression model >

What is the Anova table process?

1. Sε = A

2. R² = B

3. F-test

4. d.f._residual = C

5. P_value / Significance F < α

6. t_stat = D

What is Sε?

standard error of estimate

ŷ variation

• ~ standard deviation

What are the Sε formulas?

\sqrt{\frac{SSE}{d.f_{residual}}}

\sqrt{MSE}

What is the d.f._residual formula?

n-k-1

What is k?

independent variable number

What is the Sε & standard deviation relation?

if Sε < S_Y, small & useful Sε

What is R²?

determination coefficient

Y variation portion explained by X variation

What are the R² formulas?

r^2

\frac{SSR}{SST}

What is the F-test?

shows linear relation between Xs & Y

What is the F-test H₀?

Σ β_k = 0

What is the F-test H₁?

1+ β_k ≠ 0

What happens if no linear relation exists between 2 variables?

B₁ = 0  

What is the t_stat formula?

\frac{b_1-\beta_1}{s_{b_1}}

What is S_b1?

estimate’s standard error

What are the slope test methods?

• t_stat & t_crit

• P-value

• CI

What is the t_stat & t_crit slope test method?

1. t_stat

2. t_crit

3. IF |t_stat| > t_crit, reject

What is the t_crit Excel formula?

=T.INV.2T(α,d.f.r)

What is the P-value slope test method for simple linear regression?

= F-test

1. IF P-value / Significance F < α, reject

What is the CI slope test method?

1. IF CI doesn’t include 0, reject

What is the CI slope test formula?

b_1\pm t_{crit}s_{b_1}

What is the CI interpretation?

One is α% confident that the true relationship is within the range such that for an additional independent variable unit, the dependent variable will increase on average between LCL * 1000 & UCL * 1000.

What is the Y variance formula?

MS_{TOTAL}=S_{y}^2=\frac{SST}{n-1}

What is the F-ratio formula?

\frac{MSR}{MSE}

What are the least squares method ε_i assumptions?

• linearity

• normality

• homoscedasticity

• independent error

What is homoscedasticity?

constant variance

What is residual analysis?

examine residual

What is a residual?

data & regression difference

What is the residual formula?

e_{i}=Y_{i}-\hat{Y}

What is the standardized residual formula?

S_{r}=\frac{e_{i}}{S_{\varepsilon}}

What is the linearity assumption?

no residual plot pattern → valid

What is the residual plot x-axis?

dependent variable

What is the residual plot y-axis?

e_i

What are the normality assumption ways?

Excel…

• bell histogram

• approximately straight Normal Probability Plot

→ valid

What is the homoscedasticity assumption?

no spread change → valid

What is the homoscedasticity plot x-axis?

ŷ

What is the homoscedasticity plot y-axis?

e_i

What is the independent error assumption?

not autocorrelated in time series data → valid

What is autocorrelation?

// serially correlated

correlation over time in time series data

What are some autocorrelated examples?

• positive e_i & negative e_i runs

• e_i oscillates around 0

What is the residual analysis Excel process?

Data Analysis → Regression Analysis → Residuals / Standardized Residuals / Residual Plots / Line Fit Plots / Normal Probability Plots

What is a way to detect outliers?

IF |s_e| > 2