by omarxinyijohn
Reported first in December, 2019, the COVID-19 pandemic constitutes the largest global public health crisis in a century, with daunting health and socioeconomic challenges. In the United States alone, more than 700,000 people have lost their lives due to the public health crisis. The progress in vaccination rates, as well as the spread of the Delta variant, adds more nuance to the overall situation. In this project, we explore COVID-19 data in the United States (more specifically, the 50 states plus D.C.) using two publicly available data sets sourced from the CDC. One of these data sets includes cases and deaths data broken down by state and date, and the other includes vaccination data broken down by state and date. These data sets can be found in the reference section of this document.
We start by simply exploring the relationship between deaths and cases since the CDC began recording this type of data on January 22nd, 2020. We first visualized daily new cases versus time and daily new deaths versus time. Based on these two plots, which have almost identical shape, it appeared that daily new deaths and daily new cases are linearly related. A simple plot of daily new deaths versus daily new cases confirmed this. We then fit a linear model to this deaths versus cases data to predict how many deaths we would expect on any individual day given a certain number of cases. The linear model is given by
y = 0.01146x + 274.67437
Where x is the number of cases and y is the number of deaths. This tells us that for each additional daily case, we expect the number of daily deaths to increase on average by 0.01146. Additionally, we found that R2 = 0.673, which indicates that about 67.3% of the variation in the daily new deaths is dependent on daily new cases. Why is the R2 not any higher? Well, there could be a multitude of reasons. The most obvious reason is that we have not factored in any other variables such as vaccination rates or seasonality, but a less obvious reason is that our data is not perfectly reliable. One might notice from the daily new deaths versus daily new cases plot that there is a higher ratio of deaths to cases closer to the origin. This reflects how at the beginning of the pandemic, this ratio was higher, likely due to a combination of high susceptibility of the population dealing with the novelty of the disease and the fact that a robust testing regimen had not been developed yet. In other words, there could have been cases not counted in the data due to a lack of testing infrastructure. This problem likely persists throughout the dataset, as it is unlikely that everyone who has had COVID-19 has also been tested for it at the same time.
To further explore the relationship between deaths and cases, we also took into account the correlation and cross correlation of the two variables. Typically, when we are comparing two variables X and Y, we calculate the correlation coefficient
$$\rho = \frac{\sigma_{XY}}{\sigma_X\sigma_Y}$$
The correlation coefficient describes how one variable moves in relation to another. When dealing with time series data and both X and Y are functions of time, the time series Yt might be related to past lags of the time series Xt. In other words, if our time series is discrete (it must be), we want to determine the relationship between Xt + h and Yt for h = 0, ± 1, ± 2, ... . The cross correlation coefficient is thus given by
$$\rho = \frac{\sigma_{X_{t + h}Yt}}{\sigma_{X_{t + h}}\sigma\{Y_t}}$$
Two things to note :
When one or more Xt + h, with h negative, are predictors of Yt, it is sometimes said that x leads y
When one or more Xt + h, with h positive, are predictors of Yt, it is sometimes said that x lags y
In this case, we say cases are a predictor of deaths and cases leads deaths, so it makes sense to consider negative values of h. If we take a look at our plot of the cross correlation along with the underlying data, it appears that the highest correlation between deaths and cases is at a value of h = + 7 where correlation ≈ 0.85. However, this implies that cases in the future (plus seven days) are more related to deaths in the present than cases in the present (i.e. when h = 0), which does not make much sense in this context. One interesting thing to note is that the correlation seems to spike every seven days. In other words, there is a local maximum of correlation every seven days. If we let h = − 7 days (i.e. let the cases variable lead deaths by 7 days), then we have a local minimum of correlation equal to approximately 0.72. It is difficult to explain why the greatest correlation occurs when h is positive, but it might again be explained by the early pandemic when testing was not widely available, but determining cause of death was easy. In theory, as testing capacity increased, so did case count, which would show that cases lagged deaths for a period of time.
Another fundamental factor for analyzing COVID-19 data is vaccination rates. Due to the limitation of the two data sets in capturing the multitude of nuances involved in the pandemic (mask mandates, stay-at-home orders, differing general behaviors in different states, etc.) it is impossible to conduct a complete rigorous examination of the effectiveness of vaccination through our cursory exploration, but it is still possible to gain a modicum of insight by conducting a basic analysis of the data. We started this analysis by examining overall vaccination rates as a function of time. While this examination showed that the overall vaccination rate in the United States was only just over 55% as of early October, 2021, this hid the fact that the vaccination rate does indeed vary widely by age group. By faceting the data by age group, we saw that the 65+ age group is approaching a 90% vaccination rate, suggesting that one of the most vulnerable demographic groups in this pandemic has largely been inoculated in comparison with younger demographics. We also examined the cross correlation between daily new deaths and the percent of the population fully vaccinated. In this context, we choose vaccinations as the predictor variable. If we examine the plot and corresponding output, it would appear that vaccinations lead deaths by approximately 65 days, as this is the point where the two variables become maximally negatively correlated and h is negative. In other words, this is the point where an increase in vaccinations most strongly correlates with a decrease in deaths. This correlation is approximately − 0.313 which is not significantly strong but certainly noteworthy. Again, the caveat needs to be added that our results are only as good as our data, and given the simple nature of our analysis, it is difficult to factor in the multitude of potential variables that could also be impacting the death numbers.
The last part of our analysis involves examining the relationship between the season and deaths/cases. The COVID-19 virus is thought to spread mainly from person to person through respiratory droplets produced when an infected person coughs, sneezes, or talks. Thus, the different activities people engaged in in different seasons, as well as the weather conditions, might have had a significant impact on the spread of COVID-19. Before examining this impact, we started our analysis by defining the four seasons as:
When our data was faceted by season, we found that the winter has the highest mean and median case count, followed by the fall, summer, and spring. We also wanted to see if including the season as a predictor improved our model of deaths versus cases. To test this theory, we used the main versus interaction effects method we learned this semester. In other words, we compared the model leaving the rate of change of deaths per case the same by season (main effects) to the model where the rate of change of deaths per case changes by season. By adding the season variable as a predictor, we were able to increase the adjusted R2 value, suggesting that it makes sense to factor in seasonality into our model. However, a caveat must be added that we only have data for approximately two years of COVID-19 (two cycles of seasons), so it is possible that seasonality might not have as much of an effect given future seasonal data.
To summarize, our major findings from this cursory data analysis were the following:
One of the major limits to our data analysis is that we are viewing things on national scale and thus missing out on local state-by-state trends in our models. This is relevant because different localities have different access to vaccines, differing amounts of people taking the vaccines, differing restrictions, etc. that might lead to differing outcomes if we were to zoom in closer. For the sake of time in brevity, we limited our scope to the national scale, which blurred some of these potential distinctions. Another limitation already mentioned is that testing capacity was not very good at the beginning of the pandemic, so it is likely that the data from the early period of the pandemic is more unreliable than the rest. There is of course always the inherent limitation that cases (and perhaps even some deaths) are missed in the count as well.
Our presentation can be found here.
Surveillance Review and Response Group, 2020, Centers for Disease Control and Prevention, COVID-19 Response. COVID-19 Case Surveillance Public Data Access, Summary, and Limitations, electronic dataset, Centers for Disease Control and Prevention, https://data.cdc.gov/Case-Surveillance/United-States-COVID-19-Cases-and-Deaths-by-State-o/9mfq-cb36.
NCIRD, 2020, COVID-19 Vaccinations in the United States, Jurisdiction, electronic dataset, Centers for Disease Control and Prevention, https://data.cdc.gov/Vaccinations/COVID-19-Vaccinations-in-the-United-States-Jurisdi/unsk-b7fc.