Advanced Biostatistics                                   Quiz 5                          Name ________________________________________

April 27, 2005                         9 or 11 Total Points

 

Directions: Thoroughly, clearly and neatly answer the following two problems in the space given, showing all relevant calculations.  Unless otherwise noted, use a = 5% throughout.

 

1.       (1 + 1 + 3 + 1 = 5 or 6 points)  The Finney4 model is fit in the “First NLMixed” on p.3 in the Appendix to the ethanol (denoted “eth”) and chloral hydrate (denoted “chy”) data given in Carter et al (1988) and for which the binary response is related to the loss of righting reflex in mice.  A reduced model is fit in the “Second NLMixed” and an extended model is fit in the “Third NLMixed”, the latter model being useful just for part (d) below.

 

(a)    The q2 parameter (denoted “th2”) is estimated to be 4130.47 for these data.  Briefly and clearly interpret this parameter estimate.

 

 

 

 

 

 

(b)    The q4 parameter (denoted “th4”) is estimated to be 12.2530 for these data.  Briefly and clearly interpret this parameter estimate.

 

 

 

 

 

 

(c)    Using the best test procedure, characterize the interaction (if any) of ethanol and chloral hydrate, listing hypotheses, the test statistic, degrees of freedom (dfs), p-value, and your clear interpretation and justification.

 

Null hypothesis ________________________________________________

 

            Alternative hypothesis    ________________________________________________

 

            Calculated test statistic, distribution and dfs _________________________________________________

 

            p-value ________________________________

 

detailed conclusion

 

 

 

 

 

 

 

(d)    [Graduate students only]  The “Third NLMixed” is useful to help determine the indicated scale for these data.  Clearly interpret the results and comment on whether the scale used in the “First NLMixed” was well-chosen.

 

 

 

2.       (1 + 2 + 1 + 1 = 4 or 5 points)  The data analyzed on pp.4-5 of the Appendix, reported in Millard & Krause (2001:19), correspond to canine prostate size for 5 dogs randomized to the control group (denoted “contg”) and 5 dogs randomized to the estradiol group (denoted “estrg”), and for which relative potency of these drugs is to be assessed using a direct assay.  These data are analyzed using NLIN (p.4) and using two runs of NLMixed (p.5)

 

(a)    The best analysis for these data is (circle one):      the NLIN      the first NLMixed      the second NLMixed

 

 

(b)    Using the best analysis for these data, test whether the two drugs are equally potent.

 

Null hypothesis ________________________________________________

 

            Alternative hypothesis    ________________________________________________

 

detailed conclusion (including your justification)

 

 

 

 

 

 

 

(c)    Why are the analyses not chosen in part (a) above incorrect for these data?

 

 

 

 

 

 

 

 

 

 

 

(d)    [Graduate students only] Explain why the variance in the Second NLMixed (p.5) is written in the given form; give the justification.  What are the variances for the two groups?

 

Advanced Biostatistics                           Quiz 5 Addendum                                      27th April 2005

 

First Exercise – Programs and Output

 

First NLMixed

proc nlmixed data=one;

  parms th2=4150 th3=25 th4=10 th5=0;

  z=eth+th4*chy+th5*sqrt(th4*eth*chy);

  t=(z/th2)**th3; den=1+t; p=t/den;

  model y~binomial(n,p);

run;

 

                                     The NLMIXED Procedure

                                         Fit Statistics

                            -2 Log Likelihood                   57.7

 

                                      Parameter Estimates

  Parameter  Estimate  Std Error   DF  t Value  Pr > |t|   Alpha     Lower     Upper  Gradient

  th2         4130.47   64.8087    39    63.73    <.0001    0.05   3999.38   4261.56  -2.21E-9

  th3         20.1149    3.3433    39     6.02    <.0001    0.05   13.3524   26.8775  1.728E-8

  th4         12.2530    0.3324    39    36.86    <.0001    0.05   11.5806   12.9254  8.748E-7

  th5          0.1338   0.05704    39     2.35    0.0241    0.05   0.01846    0.2492  2.586E-6

 

Second NLMixed

proc nlmixed data=one;

  parms th2=4150 th3=25 th4=10;

  z=eth+th4*chy;

  t=(z/th2)**th3; den=1+t; p=t/den;

  model y~binomial(n,p);

run;

 

                                     The NLMIXED Procedure

                                         Fit Statistics

                            -2 Log Likelihood                   63.6

 

                                      Parameter Estimates

  Parameter  Estimate  Std Error   DF  t Value  Pr > |t|   Alpha     Lower     Upper  Gradient

  th2         4076.97   65.7330    39    62.02    <.0001    0.05   3944.01   4209.93  -1.09E-8

  th3         18.9735    3.2256    39     5.88    <.0001    0.05   12.4492   25.4978  2.037E-7

  th4         12.5168    0.3407    39    36.74    <.0001    0.05   11.8276   13.2059  -2.81E-6

 

Third NLMixed

proc nlmixed data=one;

  parms th2=4150 th3=25 th4=10 th5=0 th6=0.1;

  z=eth+th4*chy+th5*sqrt(th4*eth*chy);

  zt=(z**th6-1)/th6; th2t=(th2**th6-1)/th6;

  ex=exp(th3*(zt-th2t)); den=1+ex; p=ex/den;

  model y~binomial(n,p);

run;

 

                                     The NLMIXED Procedure

                                        Fit Statistics

                            -2 Log Likelihood                   57.5

 

                                      Parameter Estimates

                       Standard

  Parameter  Estimate     Error    DF  t Value  Pr > |t|   Alpha     Lower     Upper  Gradient

  th2         4123.50   64.8967    39    63.54    <.0001    0.05   3992.23   4254.76  -0.00169

  th3         47.1676    152.51    39     0.31    0.7588    0.05   -261.30    355.64  -0.00199

  th4         12.2378    0.3328    39    36.77    <.0001    0.05   11.5647   12.9109  0.000684

  th5          0.1327   0.05713    39     2.32    0.0256    0.05   0.01709    0.2482  0.001336

  th6         -0.1024    0.3872    39    -0.26    0.7928    0.05   -0.8856    0.6808  0.005869

 

 

Second Exercise – Programs and Output

 

NLIN run

 

data one;

  input group$ size @@;

  contg=(group='cont'); estrg=(group='estr');

datalines;

cont  1.975 cont 3.125 cont  4.433 cont  6.154 cont  4.175

estr 10.356 estr 6.313 estr 21.708 estr 12.651 estr 15.464

;

proc nlin data=one;

  parms mu1=1 rho=1;

  mean=mu1*contg+mu1*rho*estrg;

  model size=mean;

  output out=two r=r p=p;

run;

 

 

                                      The NLIN Procedure

                                    Dependent Variable size

                                      Method: Gauss-Newton

 

                                             Sum of        Mean               Approx

           Source                    DF     Squares      Square    F Value    Pr > F

           Model                      1       217.4       217.4      12.16    0.0082

           Error                      8       143.0     17.8755

           Corrected Total            9       360.4

 

                                                Approx

                  Parameter      Estimate    Std Error    Approximate 95% Confidence Limits

                  mu1              3.9724       1.8908     -0.3878      8.3326

                  rho              3.3477       1.6630     -0.4873      7.1827

 

                                Approximate Correlation Matrix

                                              mu1             rho

                              mu1       1.0000000      -0.9581651

                              rho      -0.9581651       1.0000000

 

Residual Plot

 

 

 

 

 

First NLMixed

 

 

proc nlmixed data=one;

  parms mu1=1 rho=1 sig=1;

  mean=mu1*contg+mu1*rho*estrg; var=sig*sig;

  model size~normal(mean,var);

run;

 

 

                                    The NLMIXED Procedure

 

                                         Fit Statistics

                            -2 Log Likelihood                   55.0

                            AIC (smaller is better)             61.0

                            AICC (smaller is better)            65.0

                            BIC (smaller is better)             61.9

 

                                      Parameter Estimates

                       Standard

  Parameter  Estimate     Error    DF  t Value  Pr > |t|   Alpha     Lower     Upper  Gradient

  mu1          3.9724    1.6912    10     2.35    0.0407    0.05    0.2042    7.7406   -2.2E-6

  rho          3.3477    1.4875    10     2.25    0.0481    0.05   0.03345    6.6619  -4.96E-6

  sig          3.7816    0.8456    10     4.47    0.0012    0.05    1.8975    5.6657  6.346E-7

 

 

Second NLMixed

 

 

proc nlmixed data=one;

  parms mu1=1 rho=1 sig=1;

  mean=mu1*contg+mu1*rho*estrg;

  var=sig*sig*(contg+rho*rho*estrg);

  model size~normal(mean,var);

run;

 

 

                                       The NLMIXED Procedure

 

                                         Fit Statistics

                            -2 Log Likelihood                   48.2

                            AIC (smaller is better)             54.2

                            AICC (smaller is better)            58.2

                            BIC (smaller is better)             55.1

 

                                      Parameter Estimates

                       Standard

  Parameter  Estimate     Error    DF  t Value  Pr > |t|   Alpha     Lower     Upper  Gradient

  mu1          3.9299    0.6112    10     6.43    <.0001    0.05    2.5680    5.2919  -8.08E-6

  rho          3.4209    0.7091    10     4.82    0.0007    0.05    1.8409    5.0008  -8.65E-6

  sig          1.4535    0.3582    10     4.06    0.0023    0.05    0.6553    2.2517  -9.74E-6