Reflection Paper
Baba0914
AC21620-Chapter6.pptxA Practical Approach to Analyzing Healthcare Data, Fourth Edition Chapter 6, Analyzing the Relationship between Two Variables
Susan White, PhD, RHIA, CHDA
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Learning Objectives
Illustrate appropriate inferential statistics to use for assessing the relationship between two categorical variables
Compare and contract sensitivity and specificity
Compare and contrast the two types of correlation statistics
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Categorical Variables
Descriptive Statistics
Contingency tables
Used to display and analyze the relationship between two categorical variables
Notice in table below:
20/32 = 62.5% of female patients were discharged home
10/24 = 41.7% of male patients were discharged home
Inferential Statistics
Is this just a random occurrence or is this evidence that there is a significant relationship between gender and being discharged to home?
A hypothesis test may be used to answer that question
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Example: Chi-squared Test of Independence
| Step | Response |
| 1. Determine the null and alternative hypotheses | Ho: Discharged to Home and Gender are independent H1: Discharged to Home and Gender are not independent |
| 2. Set the acceptable type I error or alpha level | The analyst is willing to accept a 5% chance or probability of rejecting the null hypothesis when it is true. Alpha = 5% or 0.05 |
| 3. Select the appropriate test statistic | Chi-squared |
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Example: Chi-squared Test of Independence
Test statistics typically compare the value observed in the sample to the null hypothesis value.
If gender and discharged home were independent, then we would expect the distribution of subjects among the four cells (Male/female x home/not home) to be uniform and not have a pattern.
In other words, the proportion of males sent home should be similar to the proportion of the females sent home if the null hypothesis were indeed true.
The basis of the chi-squared test statistic is the observed and expected frequencies in each of the table cells
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Example: Chi-squared Test of Independence
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Example: Chi-squared Test of Independence
Test statistic:
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Example: Chi-squared Test of Independence
Last two steps in hypothesis test:
Compare the test statistic to a critical value based on the alpha level and the distribution of the test statistic
Reject the null hypothesis if the test statistic is more extreme than the critical value. If not, do not reject the null hypothesis.
Chi-squared test statistic follows the Chi-squared distribution with (r-1)x(c-1) degrees of freedom. r = rows in contingency table and c = columns
Chi-squared distribution is always non-negative
Degrees of freedom define the shape
Since alpha was set to be 0.05 (5%), reject H0 if the test statistic is greater than 3.841
X2 = 2.39 which is not greater than 3.841
Do not reject H0
Conclusion: The sample data does not provide sufficient evidence to reject H0 and conclude that there is no significant relationship between gender and the likelihood being discharged to the home setting
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Sensitivity and Specificity
Measures the accuracy of predictions made by categorical variables
When using one categorical variable (smoking status) to predict another categorical variable (cancer status)
Sensitivity – proportion of sample with the indicator present and a positive test divided by the number of those with an indicator present.
Specificity – the proportion of the sample without the indicator and a negative test divided by the number of those without an indicator
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Sensitivity/Specificity Example
A health plan wishes to use accessing their patient portal as a predictor of whether or not a patient will seek care at an emergency room during the year. That is, they believe that patients that do not access the patient portal are more likely to experience an ER visit. They collected the following data based on enrollees during the previous plan year. Calculate the sensitivity and specificity of patient portal use as a predictor of ER use.
Note that the contingency table is set up so that ‘no’ for patient portal access and ‘yes’ for ER visit are in cell ‘A’ (upper left hand corner). This is because the health plan believes that patients that do not use the patient portal are MORE likely to experience an ER visit.
| ER Visit During Previous Year? | ||
| Patient Portal Access? | Yes | No |
| No | 30 | 23 |
| Yes | 15 | 86 |
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Sensitivity/Specificity Example
| ER Visit During Previous Year? | ||
| Patient Portal Access? | Yes | No |
| No | A: 30 | B: 23 |
| Yes | C: 15 | D: 86 |
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Descriptive Statistics - Correlation
Pearson’s correlation coefficient (r)
Measures the linear association between two continuous variables
Spearman’s Rho (r)
Measures the linear association between two ordinal variables or one ordinal and one continuous variable
Correlation between two variables does not imply causation – only that the two have a relationship or are ‘associated’
Be aware that correlation measures the linear association of two variables
They may be related in a non-linear way that may result in misleading values for the correlation coefficients
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Descriptive Statistics – Pearson’s Correlation Coefficient
Used for measuring the linear association between two continuous variables
Values from -1 to +1
Positive value means that both variables increase/decrease together
Example: Charges and length of stay
Negative value means that one variable increases as the other decreases
Example: Experience and time to code a medical record
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Descriptive Statistics – Pearson’s Correlation Coefficient
Example of negative correlation
More experienced coders require less time to code records – in general
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Descriptive Statistics – Pearson’s Correlation Coefficient
Example of positive correlation
Longer lengths of stay result in longer charges – in general
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Descriptive Statistics – Pearson’s Correlation Coefficient Example
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Descriptive Statistics – Spearman’s Rho Correlation Coefficient
Used for measuring the linear association between two ordinal variables or an ordinal and continuous variable
Operates on the ranks for the paired values and not the actual variable values
Typically rank ties are broken with average ranks
Values from -1 to +1
Positive value means that both variables increase/decrease together
Example: patient severity level and charges
Negative value means that one variable increases as the other decreases
Example: Grade in elementary school and time to run 100 yards
Same formula a Pearson’s r, but use ranks instead of actual values
If there are no ties in the ranks, may use (Where Di is the difference between the ranks of the ith pair of variables and n is the sample size):
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Inferential Statistics – T-test for correlations
Used to test the null hypothesis that the correlation coefficient is zero
Same formula for both Pearson’s and Spearman’s correlation coefficients
Note that the sample size in is the numerator of the test statistic
For very large samples, the test may reject the hypothesis of 0 correlation when the value of the sample correlation is not practically significant
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Inferential Statistics – T-test for correlations - Example
Test the hypothesis that the correlation between length of stay and charges in the previous example if different from zero.
Step 1: State the null and alternative hypotheses
Ho: r ≤ 0
Ha: r > 0
Note: In practice, a one sided test of significance is used for r. If the sample value is > 0, then the alternative hypothesis is ‘>0’. If the sample value is negative, then the alternative hypothesis is ‘<0’.
Step 2: Set the acceptable alpha level = 0.05
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Inferential Statistics – T-test for correlations - Example
Step 3: Determine the test statistic and calculate the value
T-test for correlations
= 0.93
Step 4: Compare the test statistic to the critical value
Use t-distribution with d.f. = n-2 = 3 and
alpha = 0.05 is 2.353
t= 4.71 > 2.353,
Step 5: Reject the null hypothesis since 4.71 > 2.353 and conclude that the correlation between LOS and charge is not zero
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Inferential Statistics Simple Linear Regression
Used to formulate a functional relationship between two continuous variables
A linear function of the independent variable (X) is estimated to predict values of the dependent variable (Y)
Slope-intercept form of a line:
Y = a + bX
a is the y-intercept
b is the slope of the line
If variables are positively correlated, the slope of the line is positive
If variables are negatively correlated, the slope of the line is negative
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Inferential Statistics Simple Linear Regression - Example
Least squares regression
Minimizes the vertical distance from each point to line
Vertical distance called the ‘error’ or ‘residual’
Least square line provides a line that comes as close as possible to all points, but may not actually intersect with any of them
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Inferential Statistics Simple Linear Regression - Example
Slope of line is 4,443
Interpretation: The expected charge increase for each additional day is $4,443
Intercept of line is $7,801
Interpretation: The expected charge with a zero day stay is $7,801
Zero stay is not realistic, but intercept gives an estimate of the fixed cost of admitting a patient while the slope represents the variable cost.
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Regression Hypothesis Tests
Two hypothesis tests are presented in this table
Ho: Intercept = 0 vs H1: Intercept ≠ 0
P-value = 0.121 > do not reject
Even though the intercept is not statistically different from zero (do not reject the null hypothesis that it is equal to zero), the intercept is typically kept in the model
Ho: Slope = 0 vs H1: Slope ≠ 0
P-value = 0.021 > reject Ho and conclude that the slope is not equal to zero
The interpretation here is that LOS gives us useful information about the charge since the slope of the regression line is non-zero
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Regression Assumptions
Residuals
Difference between the actual value of the dependent variable and the value predicted using the regression equation
The vertical (y-axis) distance from an individual point to the regression line
Must test the following assumptions regarding the residuals:
Independence
Normally distributed
Mean of zero
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