A range of values that includes the true value for the population parameter being measured. 

Confidence interval assigns an upper limit and a lower limit with an associated probability.  

Ex:  There will be a weight loss of 15-25 lb's diet supplement X with a Confidence interval of 63%

Stastical Significance

Important Questions:

Does the 95% CI include 0?

 In terms of odds ratio, does the 95% CI include 1.0?

- Precision increases

- Standard Error decreases

- The Confidence Interval will get smaller

- The probability of the outcome occurring by chance alone. 

- P < 0.5

Independent - What we're varying in the test

Dependent - What we're interested in

Ex: Look at the effect of caloric intake on weight
- Independent variable: Caloric intake
- Dependent Variable: Weight

In other words, looking at the effect of the INDEPENDENT VARIABLE on the DEPENDENT VARIABLE.

- No implied rank or rder

- Dichotomous or categorical

- Gender, ethnic background, presence of absence of X

- Categories with an implied order or rank

- Likert sclaes, beauty, military ranks, pain scale

- Constant and defined units of measurement with equal distance between values

- Celsius or Fahrenheit

- Same as interval but has an absolute zero (anchor point)

- Kelvin temperature, speed, height, mass or weight.

- Characteristics of the sample and variables

- Number of independent and dependent variables

- Data scale the variables are measured (Nominal, ordinal, interval, or ratio; Distribution).

- Study Design

- Distrubtion of the variables (Parametric vs. non-Parametric test)

- Inexpensive relative to clinical trials

- Great for topics that can't be randomized (Diet, geography, belief systems)
- Very good for preliminary research (Subject to confounders, limited variables)

- Definitive

- Causality is vastly easier to establish

- Randomization is awesome (when it works)

- More controlled

A confounder is something that confounds

- Stuns, amazes, puzzles, mystifies

- Typically refers to something unmeasured or unaccounted for in a relationship between a risk factor and an event

Ex: Coffee causes lung cancer, vaccinations cause autism.  

- The variable is normally distributed

- OR - N is large enough to consider the variable normally distributed (Central limits theorem)

- Variable is continuous (if it is discrete, it at least approximates a normal distribution)

- The variable is measured on an interval or ratio scale

- Considered to be distribution-free methods

- Sample does not have to be randomly selected

- Useful in analyzing nominal and ordinal scale data (Variable can be dichotomous)

*Dichotomous means to be divided equally into two parts.  Whatever that means.

-The basis for many statistical theories, principles and tests

- As the sample size increases the sampling distribution of the sample mean approaches the normal distribution with mean "u" and variance o²/n

As the sampling size gets large enough.........

Sampling distribution becomes almost normal regardless of shape of population and is centered on the true mean.  

This happens EVEN if the population is not normally distributed. 

- Probability statements obtained from most nonparametric statistics are exact probabilities, regardless of the shape of the population distribution from which the random sample was drawn. (ie: distribution of the variable doesn't matter)

- With small sample sizes (as small as N=6), a nonparametric test must be used.

- Treat samples made up of observation from several different populations

- Can treat data which are inherently in ranks as well as data whose seemingly numerical scores have the strength in ranks

- They are available to treat data which are classificatory (Data doesn't need to be normally distributed)

- Easier to learn and apply than parametric tests

- Losing precision/wasteful of data

- Low power

- False sense of security

- Lack of software

- Testing distributions only

- Higher-ordered interactions not dealt with

- Most of these do not concern us

- T-test

- ANOVA

- OLS

- Chi-square

- Mann-Whitney U

- Wilcoxon signed rank

- Kruskai-Wallis

-Friedman

- Logistic Regression

- GLM

- CP Hazard

- KM Curve

- Measure of population scatter

- Doesn't change (on average) with n

- A descriptive result

- Measure of estimation precision

- Gets smaller as n increases

- An analytic result

- Must be smaller than SD

- To isolate the effect of an exposure on an outcome

- To show causality (The study design and background should all build towards that goal)

- The test doesn't know what it's comparing

- Test only answers yes/no, how much, but not why or what.

- Be explicit about what you are examining

- Have a predetermined hypothesis

- Temporality

- Biological Plausability

- Strength

- Consistency

- Dose-response

- Consideration of other causes

A statistical difference is a function of the difference between means relative to variability

- A small difference between means with large variability could be due to chance.  

- The biggest difference is when there is low variability.  

- The probability that the observed difference is due to chance

- This is calculated by "difference between group means divided by variability of groups"

The "currency" of statistics - you earn a degree of freedom for every data point you collect, and you spend a degree of freedome for each parameter you estimate.  Since you usually need to spend 1 just to calculate the mean, you then are left with n-1 (total data point "n"  -1 spent on calculating the mean). 

- Corresponds to the number of sample values that can vary after certain restriction have imposed on all data values.

Use t formula to calculate a t score

If t score falls outside the fail to reject range, then reject the nul (ie, there is a difference)

- If calculated t score > score on chart, it is statistically significant (or if it falls in the shaded area on the distribution)

- Typically use alpha/2 rather than alpha for one tailed tests

- The two samples are random samples drawn from two independent populations of interest (for this type of t test)

- The measured (dependent) variable is approximately normally distributed and continous

- The variable is measured on an interval or ratio scale

- The variance of the two groups are similar

- Also called the Wilcoxon rank-sum test

- Non parametric test

- Used when data are measured (Ordinal scale, Non-normally distributed)

- Test of the equality of medians rather than means 

*A more powerful test than the student t when the assumptions of the t test are violated!!

Recap on Lecture 2:

T-tests

Paired t-tests

Non-normally distributed data

T-tests - Pairwise t-tests are more powerful but require the groups be related

For Non-normally distributed data - Mann-Whitney U (aka Wilcoxan Rank Sum)

Wilcoxon Rank Sum tests for paired data

Interpretation are the same (same set of null and alternative hypothesis)

- Stands for Analysis of Variance

- Tests the difference between the means of two different groups for normally distributed variables

- Just an extension of the t-test (an ANOVA test with only two groups is mathematically equivalent to a t-test)

Assumptions are the same as t-test

Normally distributed outtcome

Equal variances between the groups

Groups are independent

Samples are simple random samples

Samples are independent of each other

- Statistically significant ANOVA tells us that at least two of the groups differ, but not WHICH ones differ.

- Determining which groups differ requires more sophisticated analysis to correct for the problem of multiple comparisons  

- No, with 5% error per test, this could lead up to a 15% error. 

- To compare 6 groups, we would need to do 15 pair-t tests

Bonferroni Correction - Adjusts alpha by most conservative amount, divides alpha by the number of tests. 

Tukey - Adjusts P

Scheffe - Adjusts P

Know these at an overview level.  Sometimes called mutiplicity adjustments.  

Definition:

Evidence Based Medicine (EBM)

"the conscientious, explicit and judicious use of the current best evidence in making decisions about the care of individual patients."

basically use the best information to make a healthcare decision.

Definition:

clinical practice guidlines:

The best guidelines will:

1) State recommendations clearly

2)Discuss each relative alternative

3)Acknowledge possible biases and extenuating circumstances

What is a suggested approach when deciding whether to use a clinical practice guideline?

"Common sense approach" based on:

• validity
• importance
• applicability to clinical scenario

The best guidlines will:

(in regards to the quality of evidence)

1. Define admissible evidence
2. Report how it was selected and combined
3. Make key data available for your review
4. Report that they found randomized trials that link the intervention to relevant outcomes of interest

When considering if a guidline is current and not out of date the two following dates should be looked at:

1. The publication date of the most recent evidence considered.
2. The date on which the final recommendations were made.

A.  All of the treatment regimens differ

B.  I need to use a Bonferroni correction

C.  One treatment is better than the rest

D.  At least one treatment is different from the others.

E. In pairwise comparisons, no treatment will be different.  

Kruskal - Wallis - Just an extension of the Wilcoxon Sum-Rank test for 2 groups; based on ranks

Why use it?  Similar to  Mann Whitney U.  Works with medians and not means.  Does not have to be ~N. 

A.  Don't worry about it - run an ANOVA anyway

B.  Test depression for normality

C.  Use a Kruskal-Wallis (Non-parametric) ANOVA

D.  Nothing, I can't do anything with this data

E.  Run 3 nonparametric t-tests

A.  The outcome variable is normally distributed.

B.  The variance of the outcome variable is the same in all groups

C.  The groups are independent

D.  All of the above

E. None of the above.

A.  Don't worry about it - Run an ANOVA anyway

B.  Test depression for normality

C.  Use a Kruskal-Wallis (Non-parametric) ANOVA

D.  Nothing, I can't do anything with this data

E.  Run 3 nonparametric ttests

A.  Correct.  I have three means, so I have to use ANOVA.

B.  Wrong.  Blood pressure is unlikely to be normally distributed

C.  Wrong.  The variance in BP is likely to greatly differ at the three time points

D.  Correct.  It would also be OK to use three t-tests

E.  Wrong.  The samples are not independent

- For pre-post studies, a paired ttest will suffice.  1+, 1 continuous measure at two time periods.

- For more than two time periods, you need repeated measures ANOVA.  2+ groups, 1 continuosu measure at 3+ time periods.  Recall: serial paired t-tests is incorrect, because this strategy will increase your type I error.   

- Are there significant differences across time periods?

- Are there significant differences between groups

- Are there significant differenes between groups in their changes over time.  Interaction term.

A.  Repeated-measures ANOVA

B.  One-way ANOVA

C.  Two-sample t-test

D.  Paired t-test

E.  Wilcoxon sum-rank test

A.  Repeated-measures ANOVA

B.  One-way ANOVA

C.  Two-sample t-test

D. Paired t-test

E.  Wilcoxon sum-rank test

- Chi-square test allows you to compare proportions between 2 or more groups

- ANOVA for means; chi-square for proportions)

- Note:  It is not pronounced "chee" or "chai" or "X" and the "2" isa squared, not a "2."

- Used for greater than or equal to 1 groups and compares the actual number within a group to the expected number for that same group.

- Expected number is based on theory, previosu experience or comparison groups

- Address research questions related to rates, proportions, or frequencies

- Can it fit into a pie chart?

- Nominal and ordinal data

- Frequency data (pie chart)

- The measures are independent of one another

- Categorization of the variables
"Yes" or "no"
Treatment or placebo

Underweight, normal weight, or overweight

Age 0-5, 5-10, 11-15......

Many of these can also be looked at continuously.  

- Categorical data are often arranged in a table consisting of columns and rows

- Rows --> categories of one variable

- Columns --> categories of the other variable

- 2 X 2 table AKA Contingency table

- Tables representing all combinations of levels of explanatory and response variables

- Numbers in table represent COUNTS of the number of cases in each cell

- Rows and column totals are called MARGINAL counts

- Each variable has 2 levels.  

- Explanatory variable - Ex: Groups

- Response variable - Outcome, typically presence of absence of a characteristic

- Measures of association -Relative risk (prospective studies), Odds ratio (Prospective or Retrospective), Absolute Risk (Prospective Studies)

A.  Repeated-measured ANOVA

B.  One-Way ANOVA

C.  Difference in Proportions Test

D. Paired t-test

E.  Chi-square test

A.  Repeated Measured ANOVA

B.  One-way Anova

C.  Difference in proportions test

D.  Paired t-test

E. Chi-square test