Index

Abstract

Coronavirus 2019 (Covid-19) cases in Rivers State, Nigeria are on the increase day by day. It became imperative to investigate the survival rate of covid-19 patients in this state. The survival quantile regression was applied assuming right censoring to estimate the effect of   age, sex, fever, anosmia, comorbidity, and cough on the survival time of patients. The results  show that on admission into the hospital the survival time of the patients depended on the age and the presence of anosmia, comorbidity, and fever. By the mid survival period only anosmia and fever were seen to be significant but at the 75th quantile comorbidity was also seen to be  significant along with fever and anosmia. The result also shows that having fever is associated   with longer stay in the hospital based on the size of the effect at different quantiles. We also noticed that though the effect of anosmia and comorbidity were significant at the 25th and 75th quantile the sizes of the effects were minimal, but comorbidity was seen to have a bigger effect  than anosmia. Comparing the survival time of groups, the results showed that males and females have the same survival time and patients with and without comorbidity equally have the same  survival time. Patients  without fever, anosmia and cough had a shorter survival time than those that had fever, anosmia, and cough. We then concluded that fever, comorbidity, and anosmia are the major factors that affect the survival time of covid-19 patients in Rivers State, Nigeria.

Keywords: Survival quantile regression, Cox proportional Hazard model, Accelerated failure time model Log-rank test, Survival function and censoring.

Received: 3 February 2021 / Revised: 17 February 2021 / Accepted: 26 February 2021/ Published: 4 March 2021

Contribution/ Originality

This study is one of the very few studies that have investigated the effect of different covariates at different points on the distribution of the survival time of Covid-19 patients in Rivers State, Nigeria.


1. INTRODUCTION

Coronavirus 2019 (Covid-19) cases in Rivers State, Nigeria are on the increase day by day. It became imperative to investigate the survival rate of Covid-19 patients. Survival analysis is widely used in modelling and analyzing time-to-event data. Survival analysis is said to be a collection of statistical procedures for data analysis for which the outcome variable of interest is time until an event occurs, often referred to as a failure time, survival time, or event time. Survival time refers to a variable which measures the time from a particular starting time (e.g., time diagnosed of sickness) to a particular endpoint of interest, say time discharged/case resolved. The special feature about survival analysis is that in real life situations some of the event times cannot be observed due to a variety of reasons, like drop out or death before the issue is resolved. When there is sure missing information, it is termed censoring. Most researchers are often interested in estimating the survival function (or distributions of survival times), comparing survival among groups and finding the relationship between covariates and survival times. In the regression model approach for survival analysis the Cox proportional hazards model and accelerated failure time model are widely used. The introduction of quantile regression has attracted considerable interest in survival analysis. It offers a more valuable complement to the Cox proportional hazards model Cox [1] and the accelerated failure time model Buckley and James [2] in survival analysis. Quantile regression allows the covariate effects to vary at different tails of the survival time distribution. Such important heterogeneity in the population may be neglected by using the Cox model and the accelerated failure time model, hence making survival quantile regression the best alternative.

The Covid-19 virus which was first confirmed in China towards the later part of 2019 has been on the rampage across the globe. The projections of its havoc in Africa was met with a lesser impact compared to the rest of the world. Report from the World Health Organization (WHO) as at December 31st 2020, has it that there are 41,329,493 confirmed cases in America and 30,488,064 in Europe while Africa is tailing with a total number of 2,313,130 confirmed cases. Some researchers have proposed that this could be as a result of weather conditions, etc, being that as it may, we decided to investigate the effect of some factors on the survival time of covid-19 patients. Using a covid-19 dataset this paper provides a practical guide to using quantile regression in survival analysis of covid-19 patients. Thus this paper therefore focuses on application of survival quantile model to a covid-19 data. The rest of the paper is as follows; the first section presents the basic notations in survival analysis, followed by explanations of censoring, Cox proportional hazards model and accelerated failure time model, following this section is; survival quantile regression model, methodology, results and conclusions.

2. BASIC NOTATIONS IN SURVIVAL ANALYSIS

In survival analysis the function that gives the probability of the survival time occurring at exactly time t is the density function in Equation 1 ;

3. CENSORING

When event times cannot be fully observed due to a variety of reasons, data are subject to censoring which makes statistical estimation and inference for survival quantile regression more involved. Censoring is of different types, including; Right censoring, left censoring and interval censoring and censoring can be classified into fixed and random censoring. Any procedure which completely ignores censoring may give highly biased estimates Koenker [3]. Powell [4] first studied censored quantile regression with fixed censoring. Ying, et al. [5] proposed a semiparametric median regression model for random censoring. Despite the simplicity of the method in Ying, et al. [5] this procedure requires the unconditional independence of the survival time and censoring time. Relaxing the independence condition Portnoy [6] developed a novel estimating approach motivated by the classical Kaplan–Meier estimator in the one sample analysis. He provided a recursive estimator for right censored data, which only requires conditional independence between the survival time and censoring time given the covariates. Peng and Huang [7] also proposed a quantile regression for right censoring motivated by Nelson-Aalen estimator of the cumulative hazard function in univariate cases.

4. COX PROPORTIONAL HAZARDS MODEL

The Cox proportional hazards (CPH) model is one of the widely used models in survival analysis. It evaluates simultaneously the effect of the covariates on survival. Cox proportional hazards models links the hazard (the instantaneous rate of failure) to the covariates. In other words, it allows us to examine how specified variables influence the rate of a particular event happening at a particular point in time. This rate is commonly referred to as the hazard rate. The Cox model is expressed by the hazard function denoted by h(t) and it is usually given in Equation 5 as;

5. ACCELERATED FAILURE TIME MODEL

The accelerated failure time (AFT) model provides an alternative approach to the Cox proportional hazard model. The AFT models the linear relationship between log-transformation of survival times and covariates. More specifically AFT is of the form;

Koenker and Geling [8] pointed out that the CPH and AFT are limited and that they can be treated as a special case in the quantile regression model where some appropriate transformation of the survival time T is applied to make the errors independent and identically distributed.

6. SURVIVAL QUANTILE REGRESSION MODEL

Since the inception of Quantile regression in 1978, it has emerged as a powerful and natural approach to model the heterogeneous effects of covariates for a non-homogeneous population. Unlike the Cox proportional hazards model and accelerated failure time model, quantile regression models the quantile of survival time and links it to the covariates. It is robust to outliers in the data and compared to CPH and AFT models, quantile regression relaxes the proportional hazards assumption and links the whole distribution of an outcome to the covariates of interest. Quantile regression is by far a natural choice as it helps decipher the various roles each covariate plays on different quantile levels of survival. A right censoring is considered because in this research some patients died before the covid-19 is resolved/discharged and thus the exact discharge time is unobserved. Thus this work shows the usefulness of the survival quantile regression formulation for right censored outcome and transformation models, generally.

The most basic quantile regression survival model was introduced by Koenker and Geling [8].

7. THE LOG-RANK TEST

8. METHODOLOGY

In this analysis the ‘event’ considered here is ‘discharge’ from the hospital at which time covid-19 is assumed resolved. The data set for this work relates to 641 patients diagnosed with covid-19 in Rivers State Nigeria from May 2020 to August 2020. In this covid-19 research, we estimate and infer the relationship between outcomes, say, time a covid-19 patient is admitted into the hospital to the time when  he/she is discharged from the hospital (denoted by T(survival time)), and various covariates, like sex, age, fever, anosmia (absence of taste and smell), comorbidity (presence of hypertension and or biabetes) and dry-cough. The individuals admitted in the hospital and their data were available up until the end of August 2020, by which time 9 (1.4%) cases were censored while 632 (98.6%) cases were resolved and discharged from the hospital.

9. RESULTS

From the life table.1, it is seen that the marginal probability of covid-19 cases being resolved is 0.299  and the number of resolved covid-19 cases peaked between 6-7 days. The hazard rate also confirms    this by telling us that the covid-19 resolved rate was highest from the 7th to the 8th day. The probability of survival is approximately 1 from the first day of admission to the next 3 days, and  drops as the number of days increases.
The time interval is given in days. The graph shows an increasing hazard rate (hazard rate is the theoretical measure of the probability of occurrence of an event per unit time at risk, in this paper the event is covid-19 being resolved) which implies that as time of hospitalization increases and the symptoms drops (basically due to treatment) and the patient’s potential of covid-19 being resolved increases.

Table-1. Life Table.

Time
interval
N
events
dropouts
At risk
Hazard
Rate
Survival
rate
Cumulative
Hazard rate
Marginal
Probability
of event
[0,1)
353
3
1
352.5
0.00851
0.99148
0.00851
0.00851
[1,2)
349
3
3
347.5
0.00863
0.98293
0.01714
0.00856
[2,3)
343
7
1
342.5
0.02043
0.96284
0.03758
0.02009
[3,4)
335
44
1
334.5
0.13153
0.83619
0.16913
0.12665
[4,5)
290
54
0
290.0
0.18621
0.68048
0.35533
0.15570
[5,6)
236
70
0
236.0
0.29661
0.47865
0.65193
0.20184
[6,7)
166
104
0
166.0
0.62651
0.17877
1.27844
0.29987
[7,8)
62
61
1
61.5
0.99187
0.00145
2.27031
0.17732
[8, 9)
288
71
0
288.0
0.2465
0.3429
0.9683
0.1122
[9,10)
217
52
0
217.0
0.2396
0.2607
1.2080
0.0822
[10,11)
165
65
1
164.5
0.3951
0.1577
1.6031
0.1030
[11,12)
99
44
0
99.0
0.4444
0.0876
2.0476
0.0701
[12,13)
55
11
0
55.0
0.2000
0.0701
2.2476
0.0175
[13,14)
44
17
0
44.0
0.3864
0.0430
2.6339
0.0271
[14,15)
27
8
1
26.5
0.3019
0.0300
2.9358
0.0130

Table-2. Survival Quantile Regression Model Results.

t=0.25
Coefficients
Exp(coeff.)
Std error
T-value
P(>|t|)
Intercept
12.0000
16.2754e+4
84.4991
14.119
0.0000*
Sex
0.0000
1.0000
0.0000
0.6231
0.5333
Fever
2.0000
7.3891
0.0674
0.2968
0.0000*
Anosmia
-4.0000
0.0183
0.8508
-4.7013
0.0000*
Cough
0.0000
1.0000
0.0000
-1.9478
0.8456
Age
0.0000
1.0000
0.0000
2.2085
0.0027*
Comorbidity
-2.0029
0.1349
0.8441
-2.3727
0.0177*
T=0.50
Coefficients
Exp(coeff.)
Std error
T-value
P(>|t|)
Intercept
13.4056
66.3709e+4
2.1096
6.3546
0.0000*
Sex
0.0000
1.0000
0.0000
-0.08770
0.9301
Fever
3.0000
20.0855
0.5630
5.3286
0.0000*
Anosmia
-3.2650
0.0382
0.3648
-8.9497
0.0000*
Cough
-0.8594
0.4234
1.1395
-0.7542
0.4507
Age
0.0000
1.0000
0.0000
0.4697
0.6386
Comorbidity
-2.1406
0.1176
2.2658
-0.9447
0.3448
T= 0.75
Coefficients
Exp(coeff.)
Std error
T-value
P(>|t|)
Intercept
20.488
79.0360e+6
0.7232
28.3297
0.0000*
Sex
-0.4278
0.6519
0.2994
-1.4290
0.1530
Fever
2.6107
13.6086
0.2746
9.5062
0.0000*
Anosmia
-9.9084
0.0001
0.4713
-21.0225
0.0000*
Cough
0.3437
1.4101
0.2881
1.1929
0.2329
Age
0.0229
1.0231
0.0167
1.3695
0.1709
Comorbidity
-1.6177
0.1983
0.4591
-3.5234
0.0004*
T=0.95
Coefficients
Exp(coeff.)
Std error
T-value
P(>|t|)
Intercept
17.5008
39.8566e+6
1.1593
15.0960
0.0000*
Sex
0.0580
1.0597
0.5207
0.1115
0.9113
Fever
2.7852
16.2030
0.9426
2.9547
0.0031*
Anosmia
-3.2078
0.0404
2.7159
-1.1811
0.2376
Cough
-0.3498
0.7048
0.9171
-0.3814
0.7029
Age
0.0446
1.0456
0.0437
1.0191
0.3082
Comorbidity
0.5989
1.8201
0.9249
0.6475
0.5173

Figure-1. Margiinal hazard curve.

Table 2 presents the results of the survival quantile regression model in Equation 14, the  results show that at the 25th quantile all the covariates were significant except sex and cough which tells us that on admission into the hospital the survival time of the patients depended on their age and the presence of anosmia, comorbidity and fever. But by the mid survival time only anosmia and fever were seen to be significant and by the 95th quantile i.e as the length of stay in  the hospital increases the effect of anosmia disappears while the fever effect was still seen to be  significant. The exponentiated coefficients known as the hazard ratio shows the size of the effect of the covariates. The result shows that the size of the effect of fever increases the hazard by 7.9389 at the 25th quantile and peaks at the 50th quantile by 20.0855 and drops at the 75th quantile to  13.6086 but increases a bit at the 95th quantile to 16.2030, which means that having fever is associated with longer stay in the hospital. We also noticed that the effect of anosmia and  comorbidity at the 75th quantile were significant but the sizes of the effects were minimal though comorbidity has a bigger effect than anosmia.

10. RESULTS BASED ON SEX

Based on the graph both male and female patients have the same probability of survival and the  probability of survival is 0.74. The cumulative hazard plot also confirms that the survival time of both sex are similar.

Table-3. The Log-rank test results:

Note: Chisq= 0.1  on 1 degrees of freedom, p= 0.7.

The test result is not significant; therefore the probability of survival of male and female are not different, agreeing with the graph.

Table-4. The Confidence interval of survival time:

Sex
N
Events
Median
0.95 LCL
0.95 UCL
Female
171
171
7
7
8
Male
470
461
7
6
7

The table shows that female patients’ survival time was between 7-8 days with a median of 7 days while men patients’ survival time was between 6-7 days also with a median of 7 days which is the same for both.

11. RESULTS BASED ON FEVER

This test confirms what the graphs in Figure 4 & 5 shows, that there is a significant difference between the probability of survival for patients with fever and those without fever.

Table-6. The Confidence interval of survival time.

Fever
N
events
median
0.95 LCL
0.95 UCL
No fever
478
473
6
6
7
Yes fever
163
159
9
9
10

The patients that had fever were hospitalized between 9-10 days with a median of 9 days while those without fever were hospitalized between 6-7 days with a median of 6 days before the covid-19 issue was resolved. This shows that those with fever stayed longer on admission than those without fever.

12. SURVIVAL ANALYSIS BASED ON ANOSMIA

The log-rank test confirms that there is a significant difference in survival time between those that have anosmia and those that do not have anosmia.

Table-8. The Confidence interval of survival time.

Anosmia
n
Events
Median
LCL
UCL
Yes anosmia
79
79
9
7
9
No anosmia
560
551
7
6
7
Non response
2
2
16
NA
NA

The table shows that patients who had anosmia were hospitalized between 7-9 days with a median of 9 days while those without anosmia were hospitalized between 6-7 days with a median of 6 days before the Covid-19 issue was resolved. This shows that those with anosmia stayed longer on admission than those without anosmia.

13. SURVIVAL BASED ON COUGH

The log-rank test confirms that there is a significant difference in survival time between those that have cough and those that do not have cough.

Table-10. The Confidence interval of survival time.

Cough
N
Events
median
LCL
UCL
No cough
526
522
7
6
7
Yes cough
115
110
8
7
9

The patients that had cough were hospitalized between 7-9 days with a median of 8 days while those without cough were hospitalized between 6-7 days with a median of 7 days before the covid-19 issue was resolved.  This shows that those with cough stayed longer on admission than those without cough. 

14. SURVIVAL BASED ON COMORBIDITY

The log-rank test confirms that there is no significant difference in survival time between those     that have hypertension and or diabetes and those that do not have either or both.

Table-12. The Confidence interval of survival time.

Comorbidity
N
Events
median
LCL
UCL
No comorbidity
385
380
8
7
9
Yes comorbidity
208
204
7
7
8
No response
48
48
11
11
11

The patients that had hypertension and or diabetes were hospitalized between 7-8 days with a median of 7 days while those without hypertension and or diabetes were hospitalized between 7-9 days with a median of 7 days before the covid-19 issue was resolved.  This confirms the results of the log rank test that those with or without the presence of comorbidity had the same survival time. 

15. CONCLUSION

In conclusion we state that;

Funding: This study received no specific financial support.  

Competing Interests: The authors declare that they have no competing interests.

Acknowledgement: Both authors contributed equally to the conception and design of the study.

REFERENCES

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[3]          R. Koenker, Quantile regression. Econometric society monographs vol. 38. London: Cambridge U. Press, 2005.

[4]          J. L. Powell, "Censored regression quantiles," Journal of Econometrics, vol. 32, pp. 143-155, 1986.

[5]          Z. Ying, S.-H. Jung, and L.-J. Wei, "Survival analysis with median regression models," Journal of the American Statistical Association, vol. 90, pp. 178-184, 1995.

[6]          S. Portnoy, "Censored regression quantiles," Journal of the American Statistical Association, vol. 98, pp. 1001-1012, 2003.

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[8]          R. Koenker and O. Geling, "Reappraising medfly longevity: A quantile regression survival analysis," Journal of the American Statistical Association, vol. 96, pp. 458-468, 2001.

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