HANDY REFERENCE SHEET – HRP 259
HANDY REFERENCE SHEET 2 – HRP 259
Calculation Formula’s for Sample Data:
Univariate:
Sample proportion: [pic]
Sample mean: [pic] = [pic]
Sum of squares of x: [pic] [to ease computation:[pic]]
Sample variance: [pic]= [pic]= [pic]
Sample standard deviation: [pic] =[pic]= [pic]
Standard error of the sample mean: [pic]=[pic]
2. Bivariate
Sum of squares of xy: [pic] [to ease computation:[pic]]
Sample Covariance: [pic]= [pic] = [pic]
Sample Correlation: [pic]=[pic]
Hypothesis Testing
The Steps:
1. Define your hypotheses (null, alternative)
2. Specify your null distribution
3. Do an experiment
4. Calculate the p-value of what you observed
5. Reject or fail to reject (~accept) the null hypothesis
The Errors
Power=1-(
Confidence intervals (estimation)
For a mean (σ2 unknown):
[pic] [if variance known or large sample size([pic]]
For a paired difference (σ2 unknown):
[pic] [where [pic] = the within-pair difference]
For a difference in means, 2 independent samples (σ2’s unknown but roughly equal):
[pic] [pic] = [pic] or [pic]
For a proportion:
[pic]
For a difference in proportions, 2 independent samples:
[pic]
For a correlation coefficient
[pic]
For a regression coefficient:
[pic] [[pic]]
Common values of t and Z
|Confidence level |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |
|90% |1.81 |1.73 |1.70 |1.68 |1.66 |1.64 |
|95% |2.23 |2.09 |2.04 |2.01 |1.98 |1.96 |
|99% |3.17 |2.85 |2.75 |2.68 |2.63 |2.58 |
For an odds ratio:
95% confidence limits:[pic]
For a risk ratio:
95% confidence limits:[pic]
[pic]
Corresponding hypothesis tests
Test for Ho: μ= μo (σ2 unknown):
[pic]
Test for Ho: μd = 0 (σ2 unknown):
[pic]
Test for Ho: μx- μy = 0 (σ2 unknown, but roughly equal):
[pic]
Test for Ho: p = po:
[pic]
Test for Ho: p1- p2= 0:
[pic]
Test for Ho: r = 0:
[pic]
Test for: Ho: β = 0
[pic]
Corresponding sample size/power
Sample size required to test Ho: μd = 0 (paired difference ttest):
[pic]
Corresponding power for a given n:
[pic]
Smaller group sample size required to test Ho: μx – μy = 0 (two sample ttest):
(where r=ratio of larger group to smaller group)
[pic]
Corresponding power for a given n:
[pic]
Smaller group sample size required to test Ho: p1 – p2 = 0 (difference in two proportions):
(where r=ratio of larger group to smaller group)
[pic]
Corresponding power for a given n:
[pic]
Sample size required to test Ho: r = 0 (correlation/equivalent to simple linear regression):
(where r=ratio of larger group to smaller group)
[pic]
Corresponding power for a given n:
[pic]
Common values of Zpower
|Zpower: |.25 |.52 |.84 |1.28 |1.64 |2.33 |
|Power: |60% |70% |80% |90% |95% |99% |
Linear regression
Assumptions of Linear Regression
Linear regression assumes that…
1. The relationship between X and Y is linear
2. Y is distributed normally at each value of X
3. The variance of Y at every value of X is the same (homogeneity of variances)
ANOVA TABLE
| | | | | | |
|Source of variation | | |Mean Sum of Squares | | |
| |d.f. |Sum of squares | |F-statistic |p-value |
|Between |k-1 |[pic] |[pic] |[pic] |Go to |
|(k groups) | | | | |Fk-1,nk-k |
| | | | | |chart |
|Within |nk-k |[pic] |[pic] | | |
|Total variation |nk-1 |TSS=[pic] | | | |
Coefficient of Determination: [pic][pic] =[pic]
| | | | | | |
|Source of variation | |Sum of squares |Mean Sum of Squares | | |
| |d.f. | | |F-statistic |p-value |
|Model |k-1 |[pic] |[pic] |[pic] |Go to |
|(k levels of X) | | | | |Fk-1,N-k |
| | | | | |chart |
|Error |N-k |[pic] |[pic] | | |
|Total variation |N-1 |TSS=[pic] | | | |
ANOVA TABLE FOR linear regression (more general) case
Coefficient of Determination:
[pic][pic] [pic]
Probability distributions often used in statistics:
T-distribution
Given n independent observations[pic], [pic]
[pic]
The Chi-Square Distribution
[pic]; where Z~ Normal(0,1)
[pic]
The F- Distribution
Fn,m=[pic]
Summary of common statistical tests for epidemiology/clinical research:
Choice of appropriate statistical test or measure of association for various types of data by study design.
| | |
|Types of variables to be analyzed | |
| | |
| | |
| |Statistical procedure |
| |or measure of association |
|Predictor (independent) variable/s |Outcome (dependent) variable | |
| |
|Cross-sectional/case-control studies |
|Binary |Continuous |T-test* |
|Categorical |Continuous |ANOVA* |
|Continuous |Continuous |Simple linear regression |
|Multivariate |Continuous |Multiple linear regression |
|(categorical and continuous) | | |
|Categorical |Categorical |Chi-square test§ |
|Binary |Binary |Odds ratio, Mantel-Haenszel OR |
|Multivariate (categorical and |Binary |Logistic regression |
|continuous) | | |
| |
|Cohort Studies/Clinical Trials |
|Binary |Binary |Relative risk |
|Categorical |Time-to-event |Kaplan-Meier curve/ log-rank test |
|Multivariate (categorical and |Time-to-event |Cox-proportional hazards model |
|continuous) | | |
|Categorical |Continuous—repeated |Repeated-measures ANOVA |
|Multivariate (categorical and |Continuous—repeated |Mixed models for repeated measures |
|continuous) | | |
*Non-parametric tests are used when the outcome variable is clearly non-normal and sample size is small.
§Fisher’s exact test is used when the expected cells contain less than 5 subjects.
Course coverage in the HRP statistics sequence:
Choice of appropriate statistical test or measure of association for various types of data by study design.
| | |
|Types of variables to be analyzed | |
| | |
| | |
| |Statistical procedure |
| |or measure of association |
|Predictor (independent) variable/s |Outcome (dependent) variable | |
| |
|Cross-sectional/case-control studies |
|Binary |Continuous |T-test* |
|Categorical |Continuous |ANOVA* |
|Continuous |Continuous |Simple linear regression |
|Multivariate |Continuous | |
|(categorical and continuous) | |Multiple linear regression |
|Categorical |Categorical |Chi-square test§ |
|Binary |Binary |Odds ratio, Mantel-Haenszel OR |
|Multivariate (categorical and |Binary |Logistic regression |
|continuous) | | |
| |
|Cohort Studies/Clinical Trials |
|Binary |Binary |Risk ratio |
|Categorical |Time-to-event |Kaplan-Meier curve/ log-rank test |
|Multivariate (categorical and |Time-to-event |Cox-proportional hazards model |
|continuous) | |(hazard ratios) |
|Categorical |Continuous—repeated |Repeated-measures ANOVA |
|Multivariate (categorical and |Continuous—repeated |Mixed models for repeated measures |
|continuous) | | |
*Non-parametric tests are used when the outcome variable is clearly non-normal and sample size is small.
§Fisher’s exact test is used when the expected cells contain less than 5 subjects.
Corresponding SAS PROCs:
Choice of appropriate statistical test or measure of association for various types of data by study design.
| | | |
|Types of variables to be analyzed | | |
| | | |
| | | |
| |Statistical procedure |SAS PROC |
| |or measure of association | |
|Predictor |Outcome | | |
|Cross-sectional/case-control studies | |
|Binary |Continuous |T-test* |PROC TTEST |
|Categorical |Continuous |ANOVA* |PROC ANOVA |
|Continuous |Continuous |Simple linear regression |PROC REG |
|Multivariate |Continuous |Multiple linear regression |PROC GLM |
|(categorical /continuous)| | | |
|Categorical |Categorical |Chi-square test§ |PROC FREQ |
|Binary |Binary |Odds ratio, Mantel-Haenszel OR |PROC FREQ |
|Multivariate |Binary |Logistic regression |PROC LOGISTIC |
|(categorical/ continuous)| | | |
|Cohort Studies/Clinical Trials | |
|Binary |Binary |Risk ratio |PROC FREQ |
|Categorical |Time-to-event |Kaplan-Meier curve/ log-rank test |PROC LIFETEST |
|Multivariate (categorical|Time-to-event |Cox-proportional hazards model |PROC PHREG |
|and continuous) | |(hazard ratios) | |
|Categorical |Continuous—repeated |Repeated-measures ANOVA |PROC GLM |
|Multivariate (categorical|Continuous—repeated |Mixed models for repeated measures |PROC MIXED |
|and continuous) | | | |
*Non-parametric equivalents: PROC NPAR1WAY; §Fisher’s exact test: PROC FREQ, option: exact
-----------------------
[pic]
Type II Error (()
E(Çn) = n
Var(Çn) = 2n
Variance rules for correlated random variables:
Var (x+y)=Var(x)+Var(y)+2Cov(x,y); Var (x-y)=Var(x)+Var(y)-2Cov(x,y)
χn) = n
Var(χn) = 2n
Variance rules for correlated random variables:
Var (x+y)=Var(x)+Var(y)+2Cov(x,y); Var (x-y)=Var(x)+Var(y)-2Cov(x,y)
Correct
Do not reject H0
Correct
Type I error (()
Reject H0
H0 False
H0 True
True state of null hypothesis
Your Statistical Decision
HRP261
HRP262
HRP259
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