Cochran's Test

The original desktop instructions sometimes refer to buttons that open separate input windows. In this single-window web version, enter those same values directly in the text boxes or tables on the current tab. The mathematical meaning of the inputs and outputs is unchanged.

This app computes exact probabilities for Cochran's test for dichotomous 0/1 responses using finite Markov chain imbedding (FMCI).

The app can compute:
    1. exact p-value for an observed 0/1 matrix X0
    2. exact Type I error at the observed critical value
    3. exact Type II error and power under a specified H_a

The observed matrix X0 is K-by-b:

    rows    = treatments
    columns = blocks
    entries = 0 or 1 dichotomous responses

Example with K = 4 treatments and b = 8 blocks:

    X0 = [0 0 0 1 0 0 0 1;
          1 0 1 1 1 1 0 1;
          0 1 1 0 0 1 1 0;
          0 0 1 0 1 1 0 1]

No. of Treatments
    Enter K, the number of treatment rows in X0.

Enter The Matrix
    Opens a table for entering the observed matrix X0.
    Enter the number of blocks b, click Resize Table if
    needed, then enter the K-by-b 0/1 matrix.

Enter H_a prob.
    Opens a table for entering the alternative-hypothesis
    success probability matrix P.

    P must have the same size as X0, K-by-b.
    Every entry must be strictly between 0 and 1.

    P(i,j) is the success probability for treatment i in
    block j under the alternative hypothesis H_a.

After X0 is entered, the app computes:

    column totals: x_j = sum_i X0(i,j)
    row totals:    X_i. = sum_j X0(i,j)
    critical value v = sum_i X_i.^2

Exact p-value:
    p-value = P(V >= v | H0)

Type I error:
    alpha = P(V >= v | H0)

Type II error and power, if H_a probability matrix P is given:
    power = P(V >= v | H_a)
    beta  = 1 - power

Step 1. Click Clear all items.

Step 2. Enter No. of Treatments.
    Example: K = 4

Step 3. Click Enter The Matrix.
    Set number of blocks b.
    Example: b = 8
    Enter the observed 0/1 matrix X0.
    Click Save.

Step 4. Optional: click Enter H_a prob.
    Enter the K-by-b probability matrix under H_a.
    Click Save.
    This is needed only for Type II error and power.

Step 5. Click Calculate the Probability.
    The Results panel displays the p-value, Type I error,
    and, if P is entered, Type II error and power.

K = 4, b = 8

Observed matrix:

    0 0 0 1 0 0 0 1
    1 0 1 1 1 1 0 1
    0 1 1 0 0 1 1 0
    0 0 1 0 1 1 0 1

Steps:
    Enter No. of Treatments = 4
    Click Enter The Matrix
    Set b = 8 and enter the matrix above
    Click Save
    Click Calculate the Probability

Expected p-value for this example is approximately:
    0.3344727

Use the same observed matrix as Example 1.

Alternative H_a probability matrix example:

    P = [0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2;
         0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2;
         0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2;
         0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3]

Steps:
    Enter the observed matrix X0
    Click Enter H_a prob.
    Enter the probability matrix P above
    Click Save
    Click Calculate the Probability

The app displays:
    p-value under H0
    Type I error alpha under H0
    Type II error beta under H_a
    Power = 1 - beta

1. X0 must contain only 0 and 1.

1. The number of rows of X0 must equal K.

1. The H_a probability matrix P must have the same size as X0.

4. Entries of P must be strictly between 0 and 1. Do not use 0 or 1 in P.

1. If you change K, re-enter X0 and P.

Click Clear all items to clear:
    K input
    saved observed matrix X0
    saved H_a probability matrix P
    results output