CourseKata Chapter 12

Confidence Interval (2)

Mansour Abdoli, PhD

Overview / Goals

• C.I. for One Mean - Review
• C.I. for Two Means
• C.I. for Intercept & Slope
• Paiwise C.I.
• Factors Affecting C.I.

C.I. for One Mean - Review

Point Estimate \(\pm\) ME

Computing ME

Simulation

• Resample
• Compute \(\bar y\)
• Repeate many times
• Find middle C%: \[(LL, UL)\]
• ME = \(\frac{UL-LL}{2}\)

Theoretical

• Check Assumption
  • Normality or
  • Large Sample Size
• Compute SE = \(\frac{s}{\sqrt{n}}\)
• Find \(t_{\alpha/2}\) multiplier
  from \(t\sim T(df=n-1)\)
• ME = \(t_{\alpha/2} \cdot SE\)

C.I. for Two Means

Point Estimate \(\pm\) ME \[\bar y_2 - \bar y_1 \pm ME\]

Computing ME - By Simulation

Three ways to simulate a Distribution

• Shuffle \(Y\)
  • Models no-association \(\text{Null Distribution}\)
    
• Resample dataset
  • Different groups sizes
    
• Resample within groups
  • Holds the same structure
    
• Key is finding ME for the same structure

ME and Sampling Distributions

Simulation

• Resample (Find Sampling Dist.)
• Compute \(b_1=\bar y_2-\bar y_1\)
• Repeate many times
• Find middle C%: \[(LL, UL)\]
• ME = \(\frac{UL-LL}{2}\)

Computing ME - Theoretical

• Check Assumptions

• Normality or

• Large Sample Sizes

• Compute SE & DF

• Equal Variance
  • \(s_p^2 = \frac{(n_1)s_1^2+(n_2)s_2^2}{n_1+n_2-2}\)
  • \(DF = n_1+n_2-2\)
  • SE = \(s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\)

• Unequal Variance
  • \(DF \approx min(n_1-1, n_2-1)\)
  • SE = \(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\)

• \(t_{\alpha/2}\) from \(t\sim T(df=DF)\)
• ME = \(t_{\alpha/2} \cdot SE\)

Computing ME - Theoretical

‘confint()’ Computes confidence interval

C.I. for Intercept & Slope

C.I. for Intercept & Slope

Simulation

• Resample Data

• Compute \(b_0\) and \(b_1\)

• Repeate many times

• For \(\beta_0\):
  • Find middle C%: \[(LL_{b_0}, UL_{b_0})\]
  • \(ME_{b_0} = \frac{UL_{b_0}-LL_{b_0}}{2}\)
  • \(b_0\pm ME_{b_0}\)

• For \(\beta_1\):
  • Find middle C%: \[(LL_{b_1}, UL_{b_1})\]
  • \(ME_{b_1} = \frac{UL_{b_1}-LL_{b_1}}{2}\)
  • \(b_1\pm ME_{b_1}\)

C.I. for Intercept & Slope

Theoretical

• Assumptions:
  • Linear Association
  • Normal Distribution
  • Equal Variance
• \(t_{\alpha/2}\) from \(t\sim T(df=n-2)\)
• \(ME_{b_i} = t_{\alpha/2} \cdot SE_{b_i}\)

• \(b_i\) and \(SE_i\) are from the Summary table

Coefficients

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	\(b_0\)	\(SE_{b_0}\)	\(t_{b_0}=\frac{b_0}{SE_{b_0}}\)	\(p\text{-value}_{b_0}\)
Explanatory	\(b_1\)	\(SE_{b_1}\)	\(t_{b_0}=\frac{b_1}{SE_{b_1}}\)	\(p\text{-value}_{b_1}\)

C.I. for Intercept & Slope

Recalculate C.I.s:

Paiwise C.I.

Adhoc for Single Test of Multiple Predictors

Account for \(\alpha\) Inflation

• Benferroni:
  • Use \(\alpha/K\)
    • \(K\) is the number of pairwise tests
• Tukey’s HSD
  • Use range distribution

Factors Affecting C.I.

Main Ideas

• Higher confidence level \(\Rightarrow\) wider interval
• Larger sample size \(\Rightarrow\) narrower interval
• More variability \(\Rightarrow\) wider interval
• For two means: more balanced group sizes \(\Rightarrow\) smaller SE
• For regression slope: more spread in \(X\) \(\Rightarrow\) narrower interval for \(b_1\)

A Quick Demo

Wrap-Up Questions

• If we want a narrower interval, which things can we control?
• Why does asking for 99% confidence force a wider interval?
• Why does larger sample size help even when the center stays the same?
• Why is a slope estimate more precise when the \(X\) values are more spread out?