Stats Exam 4 Flashcards
Chi Square Null
Ho: There is no relationship between the two categorical variables.
Alternative
Chi Square Alternative
Ha: There is a relationship between the two categorical variables.
Assumptions of The Chi-Square Test for Independence
- The sample should be random.
- In general, the larger the sample, the more accurate and reliable the test results are. All expected counts need to be greater than 1, with at least 80% exceeding 5 to ensure reliable use of the test. Note: this rule applies only to expected frequencies. It is acceptable for an observed frequency to be 0, provided the expected frequencies meet the criterion
Linear correlations
-Have two components: direction & size
-Both described by “r”(sample) or “ρ” (rho, population)
r = Pearson’s Correlation Coefficient
Properties of linear correlation coefficient r
- Range: -1 ≤ r ≤ 1
Scale is irrelevant (based on standardized scores)
Only measures strength of linear associations
DOES NOT IMPLY CAUSALITY
r^2
r2 = proportion of the variation in y that is determined by x
interpreting r
0.5 0.9 : correlation is very strong
r = ±1.00 : correlation is perfect
Is the given r value statistically significant?
A weak correlation (small r) can be significant.
A moderate/large correlation can occur by chance alone and be statistically insignificant.
If r is NOT significant…
the best predictor of x is x_
the best predictor of y is y_
Regression line
= A “best fit line”, y = mx + b.
Residuals
variation not explained by the regression model
Least Squares Property
Linear regression produces the smallest possible sum of squares for residuals.
S.O.S. Residuals= Unexplained Variation
If no significant correlation exists, the best estimate of Y is
the MEAN of Y
F statistic
Mean Square Regression / Means Square Residual
F Test for Regression
Tells us if the regression model is statistically significant.
Multiple Regression
Bivariate regression can be extended to multivariate data
-When 2 or more independent variables may be related to a dependent variable
Advantages
- Improved predictive value (r square)
- Estimates are more precise
r^2 (R^2)
Multiple Coefficient of Determination
r2 still equals the amount of variation in one variable, explained by other predictor variables.
Adjusted Coefficient of Determination
Adding predictor variables will increase r2, even if the contribution is trivial.
The best regression equation may not have the largest r2.
For multiple regression, use an adjusted r2
k = number of predictor variables
The adjusted r square increases only if the new variables contribution is more than what would be expected by chance alone.
What independent variables to include as predictors?
1) Consider thCommon sense & practical considerations.
- Bear “Age” may be predictive, but impractical to measure.
2) e standardized coefficients
- Independent variables converted to Z scores. “Standardized coefficients” indicated relative strength influence
3) Evaluate several regression models
Choose the model with the highest adjusted r square and the fewest variables possible
Avoid multicollinear variables (head width & ear tip distance)
Choose the equation with the lowest P value (based on F statistic in ANOVA table)
χ2
χ2 = quantifies difference from expected frequencies
Small χ2 »_space; due to random variation.
Large χ2»_space; unlikely to occur by chance.
No negative values & always a one tailed test
If the observed frequencies perfectly match the expected frequencies…
you would see 0
If the observed frequencies are vastly different than the expected frequencies…
you would see a big value
chi square calculations
Sigma[(O-E)^2/E]
Chi-Square Distribution
Skewed right
χ2 = 0 to ∞
Different curve for every degree of freedom
Degrees of freedom = (rows–1)*(columns–1)
Chi Square Review
Evaluates C C relationships
Compares expected to observed frequencies
Tests of Independence
Expected = Pn = (row totalcolumn total)/total
Used to test any frequency related hypothesis
E.g.: Car accidents are 5 times more common on weekdays than on weekends.
Number Needed to Treat to prevent one case
The number of subjects we would need to treat, to prevent one case of disease
Risk
Probability for a condition/disease
Risk Ratio
a ration of two sample risks
Risk & Risk Ratio Hypotheses
H0: RR = 1.0 Ha: RR ≠ 1.0
CI for Risk Ratio
- -RR captures 1.0 Fail to reject H0,Fail to support Ha
- -RR does not include 1.0 Reject H0, support Ha
Risk Ratio Caveat
Only applies to “natural” data, that are not “case controlled”.
Includes Prospective Studies
Randomized Controlled Trials (E.g. Salk Vaccine)
The natural incidence of disease is observed
Excludes most Retrospective Studies
Case controlled studies
Experimenter decides how many cases of each condition to include
Odds Ratios can be used in these cases
Odds compare the incidence of one condition to another (not to the total)
Odd
A ratio of the incidence of one condition to it’s complimentary condition. Not a probability.
Risk vs Odds Ratio
An odds ratio is always valid
A risk ratio is valid only if the incidence of the the response variable occurs naturally.
The component risks must be valid
Often not valid in retrospective studies
Hypothesis Testing and Confidence Interval Interpretation are the same.
H0: RR = 1.0 Ha: RR ≠ 1.0
H0: OR = 1.0 Ha: OR ≠ 1.0
