This Program is used with DTL designs in the normal distribution setting. It is used to find p-values and critical values for hypothesis tests of the true mean of the selected treatment.

Cell #1

Program instructions, don’t run

Cell #2

In[1]:= Statistics NormalDistributionUsed to load a Statistical Package

Cell #3

In[2]:= ndist1 NormalDistribution0, 1;phix_ PDFndist1, xPhix_ CDFndist1, x

Out[3]=x

222

General::spell1: Possible spelling error:

new symbol name "Phi" is similar to existing symbol "phi".

Out[4]=1

2

1 Erfx2

Defines standard normal pdf and cdf functions

Ignore spelling warning

In[5]:= ClearnA, nB, u10, s, x2, z, ffz_ phiznA nBu10nA nB sPhinA znA nBx2nBnAnBs

Out[6]=

nAnBu10z2

2nAnBs2 1 ErfnAnAnBx2z2nBnAnBs22

Cell #4

Cell #4

Clears existing values of variables that may have been used in other programs. Then defines function that is proportional to the pdf of Z.

In[7]:= nA 100;

nB 100;

u10 0;

s 10;

x1 1.8881;

x2 0.9216;

y 0.7888;

zobs nAx1nBy

Out[14]= 267.69

In[15]:= ai NnA nBu10 4nA nB sbi NnA nBu10 5nA nB s

Out[15]= 565.685

Out[16]= 707.107

Cell #5

Defines actual values from relevant clinical data and parameters of interest. μ10 is set by u10 and σ is set by s.

Cell #6

Defines the initial values for the endpoints of the integral, a & b.

In[17]:= a ai

b bi

Plotfz,z, a, bOut[17]= 565.685

Out[18]= 707.107

-400 -200 200 400 600

0.02

0.04

0.06

0.08

Out[19]= Graphics

Cell #7

Creates a plot of the distribution of Z on [a, b]. If the plot indicates that the interval is not wide enough, values of a & b need to be manually set (trial and error) so that most of the distribution is captured before proceeding with additional Cells.

Make sure most of the distribution is captured.

In[20]:= cN NIntegratefz,z, a, bpz_ fzcN

Out[20]= 25.2251

Out[21]= 0.00790765 z2400001 Erf1

200184.32 z

Cell#8

Finds the normalization constant, (1/cN), and defines the pdf of Z, p[z].

In[22]:= Pz_ NIntegratept,t, a, zNIntegrate::nlim: t z is not a valid limit of integration.

NIntegrate::nlim: t z is not a valid limit of integration.

Out[22]= NIntegratept,t, a, z

Cell #9

Defines the cdf of Z, P[z]. The warnings can be ignored in version 4.

In[23]:= zguessa b2;FindRootPz 0.95,z, zguessNIntegrate::nlim: t z is not a valid limit of integration.

NIntegrate::nlim: t z is not a valid limit of integration.

NIntegrate::nlim: t z is not a valid limit of integration.

General::stop:

Further output of NIntegrate::nlim will be suppressed during this calculation.

Out[24]=z 329.779

Cell #10

This cell finds the qth quantile of the distribution, i.e. finds z such that

P[z] = q. In the example below, the 95th percentile of the distribution is found to be 329.779. This typically would be the 5% critical value for a one-sided alternative hypothesis, e.g. HA: μ10 > 0.

In[25]:= pvalue1 NIntegratepz,z, zobs, bpvalue2 1 pvalue1

pvalue3 2 Minpvalue1, pvalue2Out[25]= 0.134478

Out[26]= 0.865522

Out[27]= 0.268956

Cell #11

Finds the various p-values that are typically of interest to researchers