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It seems that most work has been done for continuous time random variables. Our grouped time models that are used for inference are chosen to relate to these we known continuous time models. We have generalized by the cox (1972) model to include main unit variability to time model.

Survival Analysis Literature

Several experimental situations are given rise to analyze time to response on observational units (survival data) using split plot in time models [

There were initially two tanks for each of the treatment combinations, the experiment was (2 × 3) factorial for treatment combinations structure.

The (2 × 3) treatment combinations were assigned to tanks in a completely random design. From this point we used (CRD) to designate this design. The experiment was carried on for (10) days and mortality was observed on daily basis―Three hundred fish were randomized to (12) tanks, (25) fishes to each tank. The (2 × 3) treatment combinations were assigned 50 that (2) tanks received each treatment.

In

Coming from Risk set TABLE interval/ q ^ i j k

As in

Assuming that tank effects increase or decrease the survivals, i.e. assuming that there is tank variability involved, since treatment combinations were applied to main units (tanks).

Also assuming that failure time (T) is a discrete random variable since time responses were grouped into intervals ( 1 , 2 , 3 , ⋯ , k ) where (k = 7) for the experiment presented. The response for discrete setting would be some function of the number of deaths or the number of survivors.

This will give us a split plot in time where subplot units are time intervals. Failure time variability will arise from the fact that (25) fish were randomly assigned to each tank.

Assuming that conditional on being in the same tank survival times of different fish are independent, then model to be considered is.

Response = μ + ∝ i + ε i j + β k + ( α β ) i k + ∂ i j k

μ = is an over all mean.

∝ i = is treatment combination (i) effect.

Acclimation Time: | One week | Two weeks | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Zinc Concentration: | Lo | Med | Hi | Lo | Med | Hi | ||||||

Tank: | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

Day Mortality: | ||||||||||||

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 1 | 2 | 3 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 3 | 0 |

3 | 5 | 7 | 7 | 10 | 12 | 10 | 9 | 4 | 12 | 9 | 12 | 12 |

4 | 7 | 4 | 9 | 7 | 7 | 8 | 4 | 4 | 5 | 3 | 3 | 7 |

5 | 1 | 2 | 0 | 5 | 4 | 3 | 0 | 0 | 3 | 2 | 2 | 2 |

6 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

7 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Acclimation Time: | One week | Two weeks | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Zinc Concentration: | Lo | Med | Hi | Lo | Med | Hi | |||||||

Tank : | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | |

Interval/ q ^ i j k : | |||||||||||||

1 | 0.960 | 0.920 | 0.880 | 0.980 | 0.960 | 0.960 | 0.980 | 0.980 | 0.960 | 0.980 | 0.880 | 0.980 | |

2 | 0.792 | 0.696 | 0.682 | 0.600 | 0.500 | 0.583 | 0.640 | 0.840 | 0.500 | 0.640 | 0.455 | 0.520 | |

3 | 0.632 | 0.950 | 0.400 | 0.533 | 0.417 | 0.420 | 0.750 | 0.810 | 0.583 | 0.813 | 0.700 | 0.962 | |

4 | 0.917 | 0.833 | 0.917 | 0.375 | 0.200 | 0.500 | 0.958 | 0.971 | 0.571 | 0.816 | 0.714 | 0.667 | |

5 | 0.955 | 0950 | 0.917 | 0.667 | 0.500 | 0.667 | 0.958 | 0.971 | 0.875 | 0.955 | 0.800 | 0.875 | |

6 | 0.955 | 0.950 | 0.917 | 0.500 | 0.500 | 0.500 | 0.958 | 0.971 | 0.875 | 0.955 | 0.875 | 0.875 | |

7 | 0.955 | 0.950 | 0.917 | 0.500 | 0.500 | 0.500 | 0.958 | 0.971 | 0.875 | 0.955 | 0.875 | 0.875 | |

Acclimation Time: | One week | Two weeks | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Zinc Concentration: | Lo | Med | Hi | Lo | Med | Hi | ||||||

Tank: | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

Interval/n_{ijk}: | ||||||||||||

1 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 |

2 | 24 | 23 | 22 | 22 | 24 | 24 | 25 | 25 | 24 | 25 | 22 | 25 |

3 | 19 | 16 | 15 | 15 | 12 | 14 | 16 | 21 | 12 | 16 | 10 | 13 |

4 | 12 | 12 | 6 | 8 | 5 | 6 | 12 | 7 | 3 | 13 | 7 | 6 |

5 | 11 | 10 | 6 | 3 | 1 | 3 | 12 | 17 | 4 | 11 | 5 | 4 |

6 | 11 | 10 | 6 | 2 | 1 | 2 | 12 | 17 | 4 | 11 | 4 | 4 |

7 | 11 | 10 | 6 | 2 | 0 | 1 | 12 | 17 | 4 | 11 | 4 | 4 |

ε i j = is main unit variability (tank variability) with:

E ( ε i j ) = 0 , E ( ε i j ε i j ) = σ E 2 for j = j ′

B_{k} is the subplot treatment or the interval effect [

Response = f ( q ^ i j k )

Two possible choices for this function that will be considered are:

f ( q ^ i j k ) = log ( − log q ^ i j k )

and f ( q ^ i j k ) = log ( q ^ i j k )

Our model for survival analysis is based on using a split plot in time model, and there for we need to consider the related literature as we seen in

log ( q i j k ) = β ′ Z i k + E ′ i j + T k where β ∈ R P , T k ∈ R &

log ( q ^ i j k ) = log ( q i j k ) + δ i j k

q ^ i j k = S i j k / η i j k

σ i j k is a random error defined by

σ i j k = log ( q ^ i j k ) − log ( q i j k )

It is a proportional hazards model is convenient, e.g. The log(−log) model is to be preferred over the large model for the two reasons [

1) using the proportional hazards model leads to work with log(−log) model specified by the equation.

log ( − log q i j k ) = β ′ X i k + E i j + log ∫ t k − 1 t k λ 0 ( u ) d u

However, using the additive form for the Hazard leads to work with log model specified by the equation.

log ( − log q i j k ) = β ′ X i k ( t k − 1 − t k ) + E i j ( t k − 1 − t k ) + [ − ∫ t k − 1 t k λ 0 ( u ) d u ]

There for inference with directly related to the parameters of the continuous time interpretation. The log(-log) model is to be preferred since β is invariant to time grouping.

2) the log model has a restricted range. q ^ i j k ’s are observed proportions and that 0 < q ^ i j k < 1 which implies that log ( q i j k ) < 0 .

From analysis not mentioned here we conclude that for E i j = 0 , the effect of the acclimation time was important in explaining the data. For the first two time intervals there was practically no difference in several rates between acclimation times of one week and two weeks. Fish under two weeks acclimation survived better than these with one week, acclimation time in the sense that the effect became greater with time. This suggests it is better to collect the data (count the number of deaths) after a period of at least three days. There was also an effect due to Zinc

Accl. | Conc. | Tank | Time | r i j k | n i j k | S i j k | σ δ i j k 2 | y i j k |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 25 | 24 | 0.747150 | −3.1985 |

1 | 1 | 1 | 2 | 5 | 24 | 19 | 0.1728 | −1.4559 |

1 | 1 | 1 | 3 | 7 | 19 | 12 | 0.19855 | −0.779 |

1 | 1 | 1 | 4 | 1 | 12 | 11 | 0.7531 | −2.4459 |

1 | 1 | 1 | 5 | 0 | 11 | 11 | 2.08315 | −3.0782 |

1 | 1 | 1 | 6 | 0 | 11 | 11 | 2.08315 | −3.0782 |

1 | 1 | 1 | 7 | 0 | 11 | 11 | 2.08315 | −3.0782 |

1 | 1 | 2 | 1 | 2 | 25 | 23 | 0.74745 | −2.4843 |

1 | 1 | 2 | 2 | 7 | 23 | 16 | 0.1728 | −1.015 |

1 | 1 | 2 | 3 | 4 | 16 | 12 | 0.19855 | −1.2459 |

1 | 1 | 2 | 4 | 2 | 12 | 10 | 0.7531 | −1.6998 |

1 | 1 | 2 | 5 | 0 | 10 | 10 | 2.08315 | −2.9702 |

1 | 1 | 2 | 6 | 0 | 10 | 10 | 2.03815 | −2.9702 |

1 | 1 | 2 | 7 | 0 | 10 | 10 | 2.08315 | −2.9702 |

1 | 2 | 1 | 1 | 3 | 25 | 22 | 1.1878 | −2.057 |

1 | 2 | 1 | 2 | 7 | 22 | 15 | 0.12345 | −0.9604 |

1 | 2 | 1 | 3 | 9 | 15 | 6 | 0.1333 | −0.0874 |

1 | 2 | 1 | 4 | 0 | 6 | 6 | 1.1138 | −2.4459 |

1 | 2 | 1 | 5 | 0 | 6 | 6 | 1.5129 | −2.4459 |

1 | 2 | 1 | 6 | 0 | 6 | 6 | 2.012 | −2.4459 |

1 | 2 | 1 | 7 | 0 | 6 | 6 | 1.5258 | −2.4459 |

1 | 2 | 2 | 1 | 0 | 25 | 25 | 1.1878 | −3.9019 |

1 | 2 | 2 | 2 | 10 | 25 | 15 | 012345 | −0.6717 |

1 | 2 | 2 | 3 | 7 | 15 | 8 | 0.1333 | −0.4633 |

1 | 2 | 2 | 4 | 5 | 8 | 3 | 1.1138 | −0.0194 |

1 | 2 | 2 | 5 | 1 | 3 | 2 | 1.5129 | −0.904 |

1 | 2 | 2 | 6 | 0 | 2 | 2 | 2.012 | −1.2459 |

1 | 2 | 2 | 7 | 1 | 2 | 1 | 1.5258 | −0.3665 |

1 | 3 | 1 | 1 | 1 | 25 | 24 | 0.9804 | −3.1985 |

1 | 3 | 1 | 2 | 12 | 24 | 12 | 0.0946 | −0.3665 |

1 | 3 | 1 | 3 | 7 | 12 | 5 | 0.1425 | 0.1339 |

1 | 3 | 1 | 4 | 4 | 5 | 1 | 0.3279 | 0.4759 |

1 | 3 | 1 | 5 | 0 | 1 | 1 | 1.5479 | −0.3665 |

1 | 3 | 1 | 6 | 1 | 1 | 0 | 1.5609 | −0.3665 |

1 | 3 | 1 | 7 | 0 | 0 | 0 | 2.0812 | −0.3665 |

1 | 3 | 2 | 1 | 1 | 25 | 24 | 0.9804 | −3.1985 |

1 | 3 | 2 | 2 | 10 | 24 | 14 | 0.0946 | −0.617 |
---|---|---|---|---|---|---|---|---|

1 | 3 | 2 | 3 | 8 | 14 | 6 | 0.1425 | −0.1669 |

1 | 3 | 2 | 4 | 3 | 6 | 3 | 0.3279 | −0.3665 |

1 | 3 | 2 | 5 | 1 | 3 | 2 | 1.5479 | −0.904 |

1 | 3 | 2 | 6 | 1 | 2 | 1 | 1.5609 | −0.3665 |

1 | 3 | 2 | 7 | 0 | 1 | 1 | 2.0812 | −0.3665 |

−1 | 1 | 1 | 1 | 0 | 25 | 25 | 2.0408 | −3.9019 |

−1 | 1 | 1 | 2 | 9 | 25 | 16 | 0.1818 | −0.8068 |

−1 | 1 | 1 | 3 | 4 | 16 | 12 | 0.2516 | −1.2459 |

−1 | 1 | 1 | 4 | 0 | 12 | 12 | 1.9909 | −3.1487 |

−1 | 1 | 1 | 5 | 0 | 12 | 12 | 1.9909 | −3.1487 |

−1 | 1 | 1 | 6 | 0 | 12 | 12 | 1.9909 | −3.1487 |

−1 | 1 | 1 | 7 | 0 | 12 | 12 | 1.9909 | −3.1487 |

−1 | 1 | 2 | 1 | 0 | 25 | 25 | 2.0408 | −3.9019 |

−1 | 1 | 2 | 2 | 4 | 25 | 21 | 0.1818 | −17467 |

−1 | 1 | 2 | 3 | 4 | 21 | 17 | 0.2516 | −1.5572 |

−1 | 1 | 2 | 4 | 0 | 17 | 17 | 1.9909 | −3.5258 |

1 | 1 | 2 | 5 | 0 | 17 | 17 | 1.9909 | −3.5258 |

−1 | 1 | 2 | 6 | 0 | 17 | 17 | 1.9909 | −3.5258 |

−1 | 1 | 2 | 7 | 0 | 17 | 17 | 1.9909 | −3.5258 |

−1 | 2 | 1 | 1 | 1 | 25 | 24 | 1.5106 | −3.1985 |

−1 | 2 | 1 | 2 | 12 | 24 | 12 | 0.0999 | −0.3665 |

−1 | 2 | 1 | 3 | 5 | 12 | 7 | 0.27 | −0.617 |

−1 | 2 | 1 | 4 | 3 | 7 | 4 | 0.421 | −0.5792 |

−1 | 2 | 1 | 5 | 0 | 4 | 4 | 2.0231 | −2.0134 |

−1 | 2 | 1 | 6 | 0 | 4 | 4 | 2.0231 | −2.0134 |

−1 | 2 | 1 | 7 | 0 | 4 | 4 | 2.0231 | −2.0134 |

−1 | 2 | 2 | 1 | 0 | 25 | 25 | 1.5106 | −3.9019 |

−1 | 2 | 2 | 2 | 9 | 25 | 16 | 0.0966 | −0.8068 |

−1 | 2 | 2 | 3 | 3 | 16 | 13 | 0.27 | −1.5749 |

−1 | 2 | 2 | 4 | 2 | 13 | 11 | 0.421 | −1.7883 |

−1 | 2 | 2 | 5 | 0 | 11 | 11 | 2.0231 | −3.0782 |

−1 | 2 | 2 | 6 | 0 | 11 | 11 | 2.0231 | −3.0782 |

−1 | 2 | 2 | 7 | 0 | 11 | 11 | 2.0231 | −3.0782 |

−1 | 3 | 1 | 1 | 3 | 25 | 22 | 1.1872 | −2.057 |

−1 | 3 | 1 | 2 | 12 | 22 | 10 | 0.0871 | −0.2389 |

−1 | 3 | 1 | 3 | 3 | 10 | 7 | 0.2436 | −1.0309 |

−1 | 3 | 1 | 4 | 2 | 7 | 5 | 0.5058 | −1.0861 |
---|---|---|---|---|---|---|---|---|

−1 | 3 | 1 | 5 | 1 | 5 | 4 | 1.5052 | −1.4999 |

−1 | 3 | 1 | 6 | 0 | 4 | 4 | 2.0064 | −2.0134 |

−1 | 3 | 1 | 7 | 0 | 4 | 4 | 2.0064 | −2.0134 |

−1 | 3 | 2 | 1 | 0 | 25 | 25 | 1.1877 | −3.9019 |

−1 | 3 | 2 | 2 | 12 | 25 | 13 | 0.0871 | −0.4248 |

−1 | 3 | 2 | 3 | 2 | 13 | 6 | 0.2346 | −0.2585 |

−1 | 3 | 2 | 4 | 2 | 6 | 4 | 0.5058 | −0.904 |

−1 | 3 | 2 | 5 | 0 | 4 | 4 | 1.5052 | −2.0134 |

−1 | 3 | 2 | 6 | 0 | 4 | 4 | 2.0064 | −2.0134 |

−1 | 3 | 2 | 7 | 0 | 4 | 4 | 2.0064 | −2.0134 |

concentration which indicates that fish survives better with low levels of Zinc concentration than for higher levels.

The authors declare no conflicts of interest regarding the publication of this paper.

Khadher, H.I. and Sadiq, K.M. (2020) The Interaction of Time with Split-Plot Using Survival Function. Open Access Library Journal, 7: e6179. https://doi.org/10.4236/oalib.1106179