Multi-study Analysis Of Survival Data For Bovine Respiratory Disease

Reporter: Chao ‘Charlie’ Huang

Project presentation

OUTLINE• 1. Introduction– Bovine Respiratory Disease– Survival analysis– Meta-analysis– Statistical models combining multi-study– Arends’ multivariate random-effects model

• 2. Methodology – Data manipulation – Modeling

• 3. Results and discussion– No covariates method– Covariate method

• 4. Conclusion

1. INTRODUCTION• 1.1 Bovine Respiratory Disease (BRD) – a severe cattle disease– coughing, fever, dehydration and death– accounting for “approximately 75 percent of

feedlot morbidity and 50 percent to 70 percent of all feedlot deaths” in the United States (Stotts 2010).

BRD occurrence

Clinical diagnosis ( temperature,

haptoglobin, etc)

Survival analysis

Generalized linear model

Type of predictor variable Type of response variable

Censor?

Linear regression Categorical or continuous Normally distributed No

Logistic regression Categorical or continuous Binary No

Survival analysis Categorical or continuous (maybe time-dependent)

Binary Allowed

The table is modified based on Brian F. Gage, 2004

1.2 Survival analysis– models time-to-event data– censoring– incomplete observation due to death, withdrawal,

etc

– time-dependent covariates

h(t) = P{ t < T < (t + Δt) | T >t}h(t) = P{ t < T < (t + Δt) | T >t}

S(t) = P{T > t} S(t) = P{T > t}

:

ˆ( ) [1 ]i

j

j t tj

dS t

n

0 1 1( ) ( ) exp ...i i k ikh t t x x

• BRD Data from OSU Animal Science Department– Study I• 137 cattle; 21 days; covariates(reticular temperature,

haptoglobin, etc)

– Study II• 265 cattle; 42 days; covariates(rectal temperature,

haptoglobin, etc)

– Study III• 347 cattle; 56 days

• Using Study I and II, Li (2009) finished survival analysis with Kaplan-Meier method and Cox's proportional hazards regression. – Overall “nearly half of the sick animals developed the

disease in the first 7 days after arrival” and “when temperature is higher, the hazard of developing BRD is higher for both data sets”.

– “when the haptoglobin level is higher, the hazard for developing BRD also increases” for Study I, and “the two coefficients, temperature and the interaction between temperature and time, are significant” for Study II.

• Next step– Increased sample size more power– How about we combine the three studies

together?

1.3 Meta-analysis – a statistical method to combine several studies’

results targeting the same or similar hypotheses– controls between-study variation– increases statistical power

Meta-analysis of the effects of psychosocial interventions on survival time in cancer patients

• An example

• However, our data – Has messy structure

• Missing or invalid variable • Different duration

– Is observational data • No randomization• No treatment vs. treatment

• If we cannot use the traditional meta-analysis, how can we combine these three studies?

1.4 Statistical models combining multi-study

• Iterative generalized least-squares

• 1.5 Arends’ multivariate random-effects model

ˆln( ln( ))i i i i iS X Z b

Survival proportion estimated by survival analysis methods

Parameter vector of fixed effects

Parameter vector of random effects

Coefficient and covariance are estimated by iterative generalized linear regression

2. METHODOLOGY

• Data transformation • Study I

• Reticular temperature (RETT) rectal temperature (RECT) • RECT=15.88 + 0.587*RETT by

Bewley et al. (2008)• Data cleaning

• Study I• 137 animals 129 animals

• Study III • 347 animals 230 animals

2. METHODOLOGY

No covariates methodNo covariates method Covariate method Covariate method

3. RESULTS AND DISCUSSION• 3.1 No covariates method

Time

3. RESULTS AND DISCUSSION• 3.1 No covariates method

3. RESULTS AND DISCUSSION• 3.1 No covariates method

A

B

3. RESULTS AND DISCUSSION• 3.1 No covariates method

20 1 2

0 1

ˆln( ln( )) ln( ) [ln( )] (7)

ˆln( ln( )) ln( ) (6)

ij i ij

ij i ij

S day day b

S day b

The straight line model in

equation (6) The quadratic curve model in

equation (7)

Estimate Standard

error P-value

Estimate Standard

error P-value

Regression coefficient

β0 or intercept -1.9249 0.1406 0.0053 -2.4587 0.1146 0.0022 β1 for ln(day) 0.6421 0.0299 <.0001 1.3285 0.05496 <.0001 β2 for [ln(day)]2 -0.1690 0.01300 <.0001

Covariance parameter

Studies 0.0430 0.02983 Residual 0.0499 0.01381

The straight line model in

equation (6) The quadratic curve model in

equation (7)

-2 Res Log Likelihood 3.5 -73.6 AIC 7.5 -69.6

AICC 7.7 -69.4 BIC 5.7 -71.4

3. RESULTS AND DISCUSSION• 3.1 No covariates method

Study-specific result Combined result After the model in equation (6)

3. RESULTS AND DISCUSSION• 3.1 No covariates method

Study-specific result Combined result After the model in equation (7)

3. RESULTS AND DISCUSSION• 3.2 Covariate method

TimeTemperature

3. RESULTS AND DISCUSSION• 3.2 Covariate method

A B

C

D

Study I

Study II

Survival proportion 95% confidence interval

3. RESULTS AND DISCUSSION• 3.2 Covariate method

3. RESULTS AND DISCUSSION• 3.2 Covariate method

The selected fixed effect temperature, ln(day), [ln(day)]2

3. RESULTS AND DISCUSSION• 3.2 Covariate method

20 1 2 3

ˆln( ln( )) ln( ) [ln( )] (8)ij i ijS day temperature day b

The multivariate random-effects model in equation (8)

EstimateStandard

errorP-value

Regression coefficient β0 or intercept -32.7670 0.4943 0.0096

β1 for ln(day) 1.5148 0.0114 <.0001

β2 for temperature 0.7610 0.0027 <.0001

β3 for [ln(day)]2 -0.2024 0.0029 <.0001

Covariance parameterStudies 0.4655

Residual 0.0119

3. RESULTS AND DISCUSSION• 3.2 Covariate method

A

B

C

D

Study-specific results Survival proportion 95% confidence interval

Study I

Study II

3. RESULTS AND DISCUSSION• 3.2 Covariate method

3. RESULTS AND DISCUSSION• 3.2 Covariate method

Survival proportion 95% confidence interval Combined result

4. CONCLUSION• Strength – Handles the observational data– Simple and robust – Easy to be programmed in SAS®

• Weakness – Not a real survival curve– Random effects have the normal distributions– Over-fitting may occur– Journal papers?

• Future Improvement– ln(-ln) transformation• Regression splines, fractional polynomials, etc.• Simulation test may decide the best transformation

– Normal distribution assumption• A gamma distribution by Fiocco, Putter and van

Houwelingen (2009)