A frailty model approach to the randomized block design

Introduction

In survival analysis the problem of heterogeneity is dealt by incorporating frailty random variable. The first univariate frailty model was suggested by Beard (1959) [2] , considering different mortality models. The same model was independently suggested by Vaupel (1979) [4] and Lancaster (1979) [4] . Beard (1959) [2] used longevity factor instead of the term frailty and later on the term frailty was introduced by Vaupel (1979) [4] in the univariate case. We observe that same concept can be incorporated in other statistical studies suitable to solve some seemingly mysterious problems. Also Ashok Shanubhogue and Nitiraj Shete (2018) [1] , discussed the analysis of frailty model approach completely randomized design. In this research paper is we have used frailty model approach to analyze the data based on RBD. We have found many situations where the experimental units within the blocks, though homogeneous will have hidden source of variability due to environment, genetic factor etc. Thus it becomes necessary to incorporate these hidden sources of variability in the model. One way is to consider the hidden variability is explained by a common unknown random variable Z which has expected value one. The random variable Z which is used as variance modifier is known frailty random variable.

Proposed model for randomized block design (RBD)

In RBD each treatment replicated exactly once in each block. Therefore, for RBD no. of blocks (b) equals to no. of replications (r) and no. of treatments (v) equals to size of the block (k). Let ith treatment be replicated r (=b) times (i=1, 2, 3…, ν); n=vr the total no. of observation. The linear model assuming various effects to be additive becomes.

Where, yij be the yield or response in j th block receiving i th treatment µ be the general mean effect. α i be the effect due to i th treatment. β j be the effect due to j th block. ε ij be the error effect due to chance.

We assume that i. The various effects are additive in nature ii. ε ij are i.i.d. N(0, 𝜎 𝑒 2 )

Let us consider i.i.d continuous frailty random variable Z ij associated with (i, j) th experimental unit. We assume that ε ij |z ij ~N(0, (

We further assume that the distribution of Zij as standard exponential. That is,

Then, using ( 2) and ( 3), the joint distribution of Y ij and Z ij is,

Integrating above with respect to Z ij , we get,

Using ( 4) and ( 5) we get the following conditional distribution of Z ij given Y ij (6) Therefore, By solving above integral,

It can be easily seen that

Maximum Likelihood Estimates

From the joint distribution given in equation ( 4), the likelihood function is given by,

Enter the values of y ij for all i=1,2,3,…,v and j=1,2,3,…,b 2. Initially consider all z ij = 1 for all (i, j). Also compute z i. and z j. as the i th treatment and j th block total for all (i, j) and z as the sum of all z ij . 3. Use values of y ij and z ij to obtain maximum likelihood estimates of the model parameters α, µ, β and ϭ by using the equations given in ( 9), ( 10), ( 11) and (12).

Construction of ANOVA Table

Let us consider the linear model assumed in equation ( 1 Our derivation, matches with the approach followed in general least square theory discussed in Rao (2001) [6] .

Therefore, the sum of squares due to error (SSE) is given by,

Using ( 14) in ( 13) and simplifying, we get, For algebraic computation, we simplify the different sum of squares given in equation ( 14) as follow,

Example: A study was conducted to know effect of four drugs on the size of the prostate (in CC). A group of 16 BPH (Benign Prostatic Hyperplasia) patients were selected for the study whose prostate size is in between (60-65 CC). Further they were divided in to the four groups (blocks) according to their BMI value, (< 20, 20-25, 25-30 and >30). After 15 days of treatment again the prostate size was measured as given in the following Table (1). To know the significant difference between treatment effects (Effect of Drug), we need to conduct analysis randomized block design. To answer the above questions, we need to carryout ANOVA provided following assumptions are valid. In this study, we can say that there is hidden source of variability related to each unit under study. This hidden variability may be the genetic factor, intellectual level (not graded is one point), background they came (place, religion), environmental condition, eating habits etc. These hidden sources of variability cause not to detect significant difference among four drugs. This needs to be extracted, so that detect signal can be rightly detected. The proposed model addresses the above situation. The estimated values for the Z ij using algorithm of E(Z ij |Y ij ) is given in the following Table (2). Using equation ( 15) and ( 16), and estimated frailty random variable Z ij , we can compute different sum of squares. Therefore, constructed ANOVA according to new criterion is given as follow. Since the p-value for drug effect in ANOVA based on frailty model approach (Table (3) is 0.04348 which is less than 0.05 (LOS), we reject the null hypothesis. Therefore, there is significant difference between the average prostate vol. (CC) obtained from four different drugs.

Conclusion

We observed from the regular approach of conducting ANOVA for the randomized block design, to know the significance of treatment effects is not able to defect the apparent differences because of heterogeneity and carrying hidden source of variability with individual observation. This fails to detect this smaller differences which lead to very important conclusions and infer a lot about drug comparisons. Developed frailty approach to the randomized block design able to detect this difference clearer and lead to conclusive statement for use of drug to reduce the prostate volume (CC). This will be very useful for the practitioners to come up with the correct idea about drug in BPH patients.

Acknowledgement

Acknowledgement: I would like to acknowledge Muljibhai Patel Urological hospital, Nadiad, Gujarat, India for continuous support during the research study.

References

1. Shanubhogue A, Shete N. A Frailty Model Approach to the Completely Random Design, PRAJÑĀ -Journal of Pure and Applied Sciences. 2018;26:47-52.
2. Lancaster T. Econometric methods for the duration of unemployment, Econometrica. 1979;47:939-956.
3. Vaupel JW, Manton KG, Stallard E. The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography. 1979;16:439-454.