Lecture 1: Regression Analysis For Public Health Studies

Front 1

Population

Back 1

a group of items or events that are of interest to a study or experiment

Front 2

Sample

Back 2

A sub-set of the population that will represent the population

Front 3

Parameter

Back 3

an unknown fixed quantity; fixed value that indexes a distribution

i.e. estimating parameters with a sample

Front 4

Variable

Back 4

random quantity, has a distribution

Front 5

probability

Back 5

chance to see a certain event with many repetitions of experiment

Front 6

Distribution

Back 6

relative occurrence or frequency of a random variable

Front 7

statistic

Back 7

a function of data, categorized as test statistic or estimator

Front 8

statistical inference

Back 8

process of drawing conclusions from data subject to random variation

Front 9

Estimate vs. Estimator

Back 9

estimate: the value that represents a sample population's point or interval estimator based on sample data

estimator: a value, either point or interval, that is used to make inferences about the population

ex. sample mean, sample median, confidence intervals

Front 10

Quantitative

Back 10

concept of amount, numerical

eg. continuous (no gaps ex. BMI, BWT), discrete (gaps, ex. integers, ER visits)

Front 11

Qualitative

Back 11

concept of attribute, category

eg. binary, multinomial, ordinal

Front 12

correlation

Back 12

measures Direction and Strength of relationship

Front 13

regression methods

Back 13

measures direction and form of relationship

Front 14

correlation coefficient

Back 14

measure of linear association between two random variables

Y <---> X

p = -1 perfect negative linear association

p = 0 no association

p = 1 perfect positive linear association

Front 15

covariance

Back 15

measure of how much two random variables change together

Front 16

sample correlation coefficient

1. Name of test?

2. What does this estimate?

3. Ranges for strength of association

Back 16

1. Pearson correlation coefficient

2. point estimator of population correlation coefficient

3. 0-.25 poor to none

0.25-0.5 fair

0.5-0.75 fair to good

0.75-1 strong to excellent

Front 17

what is 'r'? What does it measure?

Back 17

'r' represents the correlation coefficient.

It measures the strength and direction of a relationship between two random variables

Front 18

What factor(s) influence 'r' value

Back 18

outliers

Front 19

Does 'r' have units?

Back 19

No, it is independent of measurement units; any two units can be compared

Front 20

Do the variables need to be hierarchically related?

Back 20

No, they do not need to be 'predictor' and according 'response' variables

Front 21

Is 'r' adjusted for other variables?

Back 21

No, not for simple pearson correlation coefficient

Front 22

Inference on p

Back 22

need to ask questions on page 11-12

Front 23

Spearman Rank Correlation coefficient

When is this test used?

Back 23

- non-normal, non-parametric test

Use for:

1. ordinal variables

2. non-normally distributed variables

3. data with outliers

Front 24

Regression

Back 24

statistical technique for modeling the relationship between variables

Identifies relationship between MEAN value of a random variable and corresponding values of one or more independent values

Mean/avg Y <--> X

Front 25

Hills Criteria for causality

Back 25

1. strength of association

2. Dose response relationship

3. Temporality

4. Specificity

5. Consistency

6. Repeatability/Coherence

7. Biological plausibility

Front 26

1. How do you chose what regression method to use?

2. When should each of these regression methods used?

a. linear

b. logistic

c. Poisson / log-linear

d. Cox regression

Back 26

1. Type of couture variable (Y), e.g. continuous, binary, ordinal, etc.

2.

a. continuous variables

b. binary variables

c. counts (discrete variables)

d. time of event (time -> clocks -> cox)

Front 27

What is the primary purpose for regression methods?

Back 27

1. Inference: to study the association between Y and X

2. Prediction: to predict Y for a given X

Front 28

What others purposes does regression serve?

Back 28

1. To adjust for confounders

2. Evaluate interactions between different X variables regarding change in mean response

Front 29

Inference

1. What kinds of inferences can be made?

2. Describe the functions of each type.

Back 29

1. estimation and hypothesis testing

2.

Estimation: point/interval;

*interpretation of regression coefficients is important

HT: answer questions about population;

*model parameters must be interpretable for inference to be meaningful

Front 30

Prediction

Back 30

- predict response for unobserved samples

- accuracy and precision take precedence over interpretability

Front 31

What is simple linear regression (SLR)?

Back 31

one response variable (Y) vs. one explanatory variable (Covariate, X)

to answer:

Is Y linearly associated with X?

Front 32

What is the model for SLR?

Back 32

Yi = Bo + B1x1 + e

where:

Yi = response

Bo is intercept

B1 is slope

x1 is covariate

e is random error

Front 33

What are the conditions for e or 'error'?

Back 33

normal distribution with mean zero and constant variance

Front 34

What is Multiple Linear Regression?

Back 34

regression analysis for multiple covariates

Front 35

What is the model for multiple linear regression?

Back 35

Yi = Bo + B1x1 + ... + Bkxk + e

where: one response for 'k' variables

Front 36

What are the three assumptions made when comparing two population?

Back 36

1. Independence between popls

2. Normality on distribution of parameter

3. Homoscedasticity: Equal variance between popls

Front 37

Ex. Simple linear regression

Comparison of mean blood pressure between two populations, two populations in comparison are made based on weight, Y1=150 lbs and Y2=151 lbs

1. What is the response variable and the covariate?

2. How would you represent normal distribution of sample mean blood pressure for each population?

Back 37

1. response: mean blood pressure; covariate is weight

2. Y1 ~ N(ux=150, o^2), Y2 ~ N(ux=151, o^2)

Front 38

How would you represent the population regression line if we were able to collect data from all Rutgers students, obtaining the true mean blood pressure at targeted weights?

Back 38

Uylx = Bo + B1x

where:

Uylx is mean y at a given x-value (ex. mean blood pressure at 153 lbs)

Bo is intercept

B1x is slope and covariance

Front 39

Since we are unable to obtain data from the entire Rutgers community, what is the resultant model for estimating mean blood pressure?

Back 39

Yi = Uylx + ei

where:

Yi is the sample population mean blood pressure

Uylx is the population mean blood pressure

ei is unknown random error

Front 40

How would you word results indicating a good relationship between the data and the linear model?

Back 40

The mean of response variable has a linear relationship with explanatory variable x.

Front 41

What is the primary assumption about your data when you use linear regression?

Back 41

That the relationship between your response and explanatory variable is linear.

Front 42

What are the assumption for errors?

Back 42

1. Normal distribution with mean: 0 and variance o^2 (ei ~ N(0, o^2)

2. errors are independent

3. Same variance

Front 43

What are the assumptions for a response variable?

Back 43

1. Normal distribution with mean (Uylx) and variance o^2

2. Same variance (homoscedasticity)

3. Independence

Front 44

Are covariates (Xk) fixed or random?

Back 44

Both!

We are under the assumption that they are fixed for the purposes of simplifying the model.

BUT... they may, in fact, be random, due to causes such as... measurement error and natural fluctuations