Statistics: Chi-Square & Sampling Distributions

STA 6176 Biostatistics Midterm Review

Review of Topics

• Counting Data includes:
  • Binomial model, Binomial test (small sample) and Z test (large sample) for Binomial proportion (Topic 4).
  • Z test (large sample) and Fisher's exact test (small sample) for two proportions and Hypergeometric model (Topic 5).
  • Poisson model, Cl of Binomial proportion by Poisson approximation (large n, small π) (Topic 6).
  • Multinomial model, x² Test for goodness-of-fit (Topic 7).
• Categorical Data includes:
  • x² Test for association in two-way contingency table (Topic 8).
  • x² Test for trend in 2 × k table (Topic 9).

Review by Types of Data

• For one categorical variable:
  • When there are two outcomes, use the Binomial model.
  • When there are more than two outcomes, use the Multinomial model and goodness of fit.
• For two categorical variables:
  • Use the Hypergeometric model and Fisher's test in a 2 × 2 table.
  • Analyse association in r × c table.
  • Analyse trend in 2 × k table.

One Variable, Two Outcomes

• Using a Binomial model Y ~ Bin(n, π), where π = P(success).
• To test for π, the null hypothesis Ηo : π = πο, and alternative hypothesis Η₁ : π <, >, ≠ πο.
• In the Small sample (n < 50), use the Binomial Exact test for π.
• In the Moderately large sample (n < 50 and ηπο (1 – πο) ≥ 10), use the Z test for π with continuity correction.
• In the Large sample (n < 50 and ηπο (1 – πο) ≥ 100), use the Z test for π.
• If n is large, and π is small (n ≥ 20, π ≤ 0.1, and observed y ≥ 5), use Poisson Approximated CI for π.

One Variable, More Than Two Outcomes

• Using the Multinomial model
  • (N1, N2, ..., Nk) ~ Multinomial (Ν., π1, π2, ..., πκ).
• To test for goodness-of-fit:
  • The null hypothesis Ηο : πι = πι ...., πκ = πκ, and alternative hypothesis Η₁ : At least one is not equal.
  • Calculate The expected value using E(N;) = ηπ.
  • Degrees of freedom is DF = k - 1.
  • Requires a large sample.

Topic 8 Chi-Square Test for Association

Categorical Data Analysis

• Focuses on Chapter 7 Categorical Data
• Study the relationships of two categorical variables, variables can have more than two levels/categories
• Example analysis could be smoking status vs cancer status
• We count the number of occurrences under each pair of conditions and enter them into a (two-way) contingency table
  • Contingency table a generalization of a 2×2 table
• Generalized contingency table
  • r # of rows
  • c = # of columns
  • i = row index
  • j = column index
  • nij = number of occurrences in ith row and jth column

Two-Way Contingency Table

• Two-way contingency tables are a r × c contingency table.
• r represents the number of rows.
• c represents the number of columns.
• i is the index of row levels, where i = 1, 2, ..., r.
• j is the index of column levels, where j = 1, 2, ..., c.
• represents the number of occurrences in the i row level and j" column level.

Two-Way Contingency Table Example

• Example 7.1 Gastric freezing

  • A balloon was lowered into the patients stomach
  • Then coolant was placed in the balloon
  • This was to see if it would heal duodenal ulcers
  • There were 2 conditions freeze and sham
    • sham = controlled variable, everything else is the same, tube in the mouth

• Example Question , is there any difference between the treatment and control

  • Equivalently, any association between the treatments and the cause of endpoints?

Probability Model in Two-Way Contingency Table

• Taking a sample of units from the population
• Observing each unit and take values of two categorical variable ( one from the column the other from the row)
• πij = probability that the row variable takes on level i and column variable takes on level j
• The sum of all probabilities will be 1
• πi. = sumj(πij ) = P(Row == level i), sum over the columns for the rows and the probability the row equals the ith level
• π.j = sumi πij = P(Column == level j), sums over the rows for each column to get the probability that the column == jth level
• Think of Nij as a random variable conditioning the row totals col totals and total overall
• Nij bin (n., πij) so E(Nij ) = n.. π I J, or π I J = E(Nij)/N

Chi-Square Test for Association

For Row and column variables

• Ho: No association
• H1: There is association
• P(A n B) == P(A) x P(B)
• No assocation between row and column variables P (row == and column == j) == P (Row == I)) ( P Column j)
• Ho: π == (pi, )( pj,)
• H1: π != (pi, )( pj, )
• if we have a 2x2 table than Ho mean that there is no differences between the probabilities If H0 of NO ASSOCATION than πij = πi + πj implies that E(nij) approximately equal to (ni. x n.j)/ni. nj. N..is the expected count
• So at ith row and jth column we get X2 equals - to test statistic similarly to goodfit Test X2 equals (observed i–Expected)2 / expected

Chi-Square Test For Goodness of Fit

• Useful for determining if observed sample data is consistent with hypothesized distribution. If observed and expected data are close = good fit

• Hypotheses:

  • H0: Data follows a specified distribution
  • H1: Data does not follow distribution
  • Test statistic: 2 K Observed i -Experi
  • X2 = equals, equals
  • Sum at I minus equals 1

• Expected

• Sampling distribution:

  • X2 degrees of freedom (k – 1).

Multinomial Model

• It can be used when you can count the data in different multinomial models

• Trial has K outcomes where K is greater than 2. K equals the number of categories.

• P(Outcome J) = TTJ, I = 12 K

• N independent identical trials

• The number of outcomes I has multinomial distribution.

• Notation ni multinomial and pi

• the means values e(ni) = npi

Example of Multinomial Models

• AA DOMINANT aa recessive
• Two parents are a the spring to be AA to a or probability.
• Ratio1.2.1 To probability 1/4 1/2 1/4, if we consider 639 offsprings.

The mean is expected values are :

Expected to be en wonder equals, E and one equals.

Poisson Model

• Used to Model the counts of event in time or space, assuption is the of event that occurs in space or space time, should independant of area where we are modelling. Ex number of arrivals to emergency room

• Approximate binomial model when n is large , and pi is small the assuption is n greater and equal to 20, and pI less then 1. The number of diseases.

• let Y mean is a parameter of lambda. e^(-y) lambda k/k

• The observed Y is what can be used to make this model to predict what a confidence model may look. ((sqrtY-1) squared),(sqrtY+1) squared)

• The Cl for lambda and approximate Cl

• let X mean is a binomial test , we need to check it is greated and equals to 5. (square root (x)-1) squared/n (square root (x)+1 squared/n

Questions cover multinomial distributions, Chi-Square goodness-of-fit tests, null hypothesis testing, and sampling distributions. It also covers sample statistics and point estimators.