Identification of Thresholds on Population Density for Understanding Transmission of COVID‐19

Abstract Pathways of transmission of coronavirus (COVID‐19) disease in the human population are still emerging. However, empirical observations suggest that dense human settlements are the most adversely impacted, corroborating a broad consensus that human‐to‐human transmission is a key mechanism for the rapid spread of this disease. Here, using logistic regression techniques, estimates of threshold levels of population density were computed corresponding to the incidence (case counts) in the human population. Regions with population densities greater than 3,000 person per square mile in the United States have about 95% likelihood to report 43,380 number of average cumulative cases of COVID‐19. Since case numbers of COVID‐19 dynamically changed each day until 30 November 2020, ca. 4% of US counties were at 50% or higher probability to 38,232 number of COVID‐19 cases. While threshold on population density is not the sole indicator for predictability of coronavirus in human population, yet it is one of the key variables on understanding and rethinking human settlement in urban landscapes.

Odds (or odds ratio) will be 1− and can be the positive value, and the logit (logarithmic of odds) can be any positive or negative value. Therefore, a linear relationship can be assumed between the logit and X: Also, = exp ( + ) 1+exp ( + ) [4] This model is called the logistic regression model (Hosmer& Lemeshow , 2000,p.6). Occasionally the response variable Y can have more than two values (0 and 1) and follow order i.e., ordinal.
Simple logistic regression cannot be applied; hence cumulative probabilities, cumulative odds and cumulative logits are used.
For k+1 ordered categories, ( <= i) = p1 + p2 + … . . pi = ln ( 1− ( ≤ ) ) , = 1,2, … , [5] Hence, for m covariates the cumulative logistic model for ordinal response data is ( ≤ ) = α + β i1 + ⋯ + β im [6] The general cumulative logistic regression model contains many parameters as we have k model equations and one logistic coefficient β ij for each covariate combination. It signifies that the odds differ only for the intercepts α, for each cut-off category, i.e. the odds are proportional. MINITAB uses the proportional odds model where a vector of predictors, X, has a parameter β describing the effect of X on the log odds of the response in category k or below.
Minitab assumes an identical effect of x for all k -1 category, so only one coefficient is calculated for each predictor. The coefficient for the predictor indicates that for any fixed k, the estimated change in the logit of the response when predictor is at one level compared to the reference level.
Here in this study, the proportional odds model is used where response Y (COVID19 cases) is a categorical variable, grouped as low, medium, and high cases, and a vector of the predictor (population density), X, has a parameter β describing the effect of X on the log odds of the response Y in category k or below.
To determine whether the pairs are concordant or discordant, MINITAB calculates the cumulative predicted probabilities of each observation and compares these values for each pair of observations.

Concordant pairs
For pairs that include the lowest response value (in the example above, that is 1), a pair is concordant if the cumulative probability up to the lowest response value is greater for the observation with the lowest response value than for the observation with the higher response value.
For pairs with the highest response values (in the example above, pairs with 2 and 3), a pair is concordant if the cumulative probability up to 2 is greater for the observation with the response value 2 than the observation with the response value 3.

Discordant pairs
For pairs that include the lowest response value (in the example above, that is 1), a pair is discordant if the cumulative probability up to the lowest response value is greater for the observation with the higher response value than for the observation with the lower response value.
For pairs with the highest response values (in the example above, pairs with 2 and 3), a pair is discordant if the cumulative probability up to 2 is greater for the observation with the response value 3 than the observation with the response value 2.

Ties
A pair is tied if the observations have equal cumulative probabilities.
The below mentioned terms explain the key components of the fitted ordinal logistic model for testing its adequacy and subsequently for interpretation of the results

i. Link function
In general, a link function (Hosmer& Lemeshow ,2000,p.48 Where ( ) is the logit link function given by ( ) = ( 1− ) p is the event probability K=number of distinct categories of the response (three (03) in our study, where high=1;medium=2,and low=3) =constant associated with the k th distinct response category = a vector of predictor variables = a vector of coefficients associated with the predictors

ii. Coefficients and constants
Coefficients are used to examine the change in the probability of response variable due to the change in predictor variables. A positive value indicates that an increase in predictor value more likely than otherwise. Negative values signify that the last event or the event close to it is more likely than otherwise. A value close to zero signifies that predictor has a little effect on the response probability. Proportional odds model was used, and it assumes that the effects of predictor variables are the same for all categories on the logarithmic scale. It further implies that the model has different intercepts but common slopes (coefficients) among categories.
Thus, the general form of regression equation using logit function is given by If the number of categories were j, then j-1 regression equations suffice to determine the cumulative probabilities for each of the j categories as the probabilities for j th category was 1.
In Table S1, constant 1 (high cases) and constant 2 (medium cases) denote the values of 1 and 2 respectively, while the coefficient for population density is denoted by . These constants and coefficients were estimated using maximum likelihood estimation method (Hosmer & Lemeshow, 2000,p.63). is generally sufficient to reject the null hypothesis. A significance level of 0.05 is generally used which means that there is 5% risk of concluding that relationship exists between the response and model terms. In this study, p-values for each of the constants and coefficient were zero at significance level of 0.05 (Table S1) signifying that a significant association exists between the response and the terms (constants and coefficients).

vi. Odds ratio
Odds ratio is used to understand the effect of predictor on the response probability.
Mathematically defined as

= ℎ ℎ
A value greater than 1 indicates that the with the increase in predictor, first event and the events closer to the first event become more likely.
Within the context of this study, value of odds ratio is greater than 1 implies that the probability of high cases increases with the increase in population density.

vii. Log-likelihood
This metric is mainly used to compare two models to help in model selection. A value close to zero suggest that the data fits the model well (Hosmer & Lemeshow, 2000,p.9).

viii. Test of all slopes equal to zero
This test is used to determine whether at least one of the predictors in the model has a statistically significant association with the response events. Degree of Freedom (DF) is the number of coefficients of predictors used in the model. A p-value less than or equal to the significance level(usually 0.05) indicates that statistically significant association exists between the response and at least one of the predictors("Methods and formulas for Ordinal Logistic Regression -Minitab," 2019) .

ix. Deviance goodness of fit test
A statistic that indicates that how well the model fits the data. A p-value greater than 0.05 indicates good fit (Hosmer & Lemeshow, 2000,p.37). x

Goodman-Kruskal Gamma
Goodman-Kruskal Gamma is the percentage difference between concordant and discordant pairs, excluding ties.

Kendall's Tau-a
Kendall's Tau-a is the percentage difference of concordant and discordant pairs out of all possible pairs, including pairs with the same response value.
All the three above mentioned measures of association are used to compare the predictive performance of the models. They can be estimated by formulas given below ("Methods and formulas for Ordinal Logistic Regression -Minitab," 2019).