9/17/2015. Basic Statistics for the Healthcare Professional. Relax.it won t be that bad! Purpose of Statistic. Objectives

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1 Basic Statistics for the Healthcare Professional 1 F R A N K C O H E N, M B B, M P A D I R E C T O R O F A N A L Y T I C S D O C T O R S M A N A G E M E N T, LLC Purpose of Statistic 2 Provide a numerical summary of the data being analyzed. Data (n) Factual information organized for analysis. Numerical or other information represented in a form suitable for processing by computer Values from scientific experiments. Provide the basis for making inferences about the future. Provide the foundation for assessing process capability. Provide a common language to be used throughout an organization to describe processes. Relax.it won t be that bad! Objectives Identify the basic tenets of statistics and statistical theory Define mean, median and other central measurements Define standard deviation, inter-quartile ratios and other measurements of variability Explain the difference between data analysis and statistics Describe hypothesis testing and other tests of statistical significance Articulate how to build relationships through regression analysis 3 1

2 3 Degrees of Separation Measures of Location (central tendency) Mean, Median and Mode 1st statistical moment Measures of Variation (dispersion) Range, Standard deviation, Interquartile Range 2nd statistical moment Measures of Error (estimation) Standard error and confidence intervals Measures of Relationships Covariance, Correlation, Regression 4 Descriptive Statistics Descriptive statistics are numbers that are used to summarize and describe data Mean conversion factor Cost per RVU Average Collection by provider Work RVUs that define 1 FTE New office visits to initial consults Measures of central tendency include the mean, median, and mode Measures of variability include the range, variance, and standard deviation. Descriptive statistics are just descriptive and do not involve generalizing beyond the data at hand 5 Inferential Statistics Inferential statistics depends upon a sample of a population to draw (or infer) conclusions about the population as a whole Inferences are made based on central tendency or any of a number of other aspects of a distribution For example, it is not practical to review every chart for a practice so extrapolation is used to assess overpayment No given sample will represent exactly the population, so distribution techniques and sample error calculations are very important 6 2

3 Important Definitions Universe The complete set of objects included within the database in question Sample Frame A homogenous set of objects that the investigator is interested in studying Sample A subset of the population that is actually being studied Variable A characteristic of an individual or object that can have different values (as opposed to a constant) Independent variable The variable that is systematically manipulated or measured by the investigator to determine its impact on the outcome. Important Definitions Dependent variable The outcome variable of interest Data The measurements that are collected by the investigator Statistic Summary measure of a sample Parameter Summary measure of a population Attribute Data (Qualitative) Types of Data Is always binary, there are only two possible values (0, 1) Yes, No Go, No go Pass/Fail Variable Data (Quantitative) Variables are properties or characteristics of some event, object, or person that can take on different values or amounts (as opposed to constants such as π that do not vary). Independent Variables Variables that are manipulated by the experimenter Dependent Variables A variable that measures the experimental outcome 9 3

4 Discrete Variables 10 Discrete variables are whole numbers (count numbers) that do not pass through the space between each number. The number of patients seen in a single day The number of surgical procedures reported by a provider The number of codes reported for the practice The number of different specialties The number of charts with coding errors Continuous Variables Continuous variables are real numbers that can occupy and infinite amount of space between discreet variables Frequency distribution of E/M codes within a category Total number of calculated FTEs in a practice Minutes per work RVU Ratio of initial consults to new office visits Cost per RVU Charge per Hour 11 Frequency Distribution 12 W E L C O M E T O T H E F A M I L Y! 4

5 Frequency Distributions A frequency distribution shows the number of observations falling into each of several ranges of values Frequency distributions are portrayed as frequency tables, histograms, or polygons Frequency distributions can show either the actual number of observations falling in each range or the percentage of observations From the frequency distribution table, you can calculate the mean, median, mode, and range 13 Normal Distribution In many natural processes, random variation conforms to the normal distribution Characteristics Symmetric, Unimodal, Extends to +/- infinity Completely described by two parameters Mean and Standard deviation % of the data will fall within +/- 1 standard deviation % of the data will fall within +/- 2 standard deviations % of the data will fall within +/- 3 standard deviations 14 Normal Distribution - Illustrated

6 Properties of the Normal Distribution 1. It is bell-shaped 2. The mean, median and mode are equal and located in the center of the distribution 3. It is unimodal (has only one mode) 4. The curve is symmetric about the mean 5. The curve is continuous (for each value of x there is a corresponding value of y) 6. The curve never touches the x-axis (goes to infinity) 7. The total area under the curve is The area under the curve that lies 1, 2 and 3 STD is equivalent to 68%, 95% and 99.7% respectively 16 Same Mean, Different Standard Deviations 17 Different Mean, Same Standard Deviation 18 6

7 Different Mean, Different Standard Deviation 19 Skew Looking at the Curve Skew measures the degree to which the distribution is biased (or skewed) right or left of normal expectation Kurtosis (it s not a disease) Beyond skewness, kurtosis tells us when our distribution may have high or low variance, even if normal. 20 Skewness the 3 rd Moment 21 The third moment, is used to define the skewness of a distribution: Skewness is a measure of the symmetry of the shape of a distribution. If a distribution is symmetric, the skewness will be zero. If there is a long tail in the positive direction, skewness will be positive, while if there is a long tail in the negative direction, skewness will be negative. 7

8 Percent Distribution Mean Median Anderson-Darling Normality Test A-Squared 0.13 P-Value Mean StDev Variance Skewness Kurtosis N Minimum st Quartile Median rd Quartile Maximum % Confidence Interval for Mean % Confidence Interval for Median % Confidence Interval for StDev Mean Median Individual standard deviations are used to calculate the intervals Anderson-Darling Normality Test A-Squared P-Value <0.005 Mean StDev Variance Skewness Kurtosis N 2507 Minimum st Quartile Median rd Quartile Maximum % Confidence Interval for Mean % Confidence Interval for Median % Confidence Interval for StDev /17/2015 Kurtosis The 4 th Moment The fourth moment, is used to define the kurtosis of a 22 distribution: Kurtosis is a measure of the flatness or peakedness of a distribution. Flat-looking distributions are referred to as platykurtic, while peaked distributions are referred to as leptokurtic. A kurtosis value of 3 represents a normally distributed data set. <3 approaches platykurtic while > 3 approaches leptokurtic. Normal or Not? 23 Normal Not Normal Summary Report for Data Summary Report for Frequency 95% Confidence Intervals 95% Confidence Intervals Benford s Distribution Benford's Distribution The distribution of first digits in any series of naturally occurring numbers, according to Benford's law. Each bar represents a digit, and the height of the bar is the percentage of numbers that start with that digit First Digit

9 Percent Distribution 9/17/2015 Even We Are Obliged to Follow Benford 25 Benford's Distribution Benford Powers of 2 Area Population First Digit Weighted average charges for 11,066 individual procedure codes Taken from the 100% Medicare claims database Measures of Position and Central Tendency 26 M E A N, M E D I A N, M O D E, P E R C E N T I L E S Measures of Central Tendency In the study of statistics there are many measurements of central tendency. The three most common are the mean, median, and mode. These metrics are used to identify the approximate location of the center of the data 9

10 Mean The mean, or arithmetic average, is found by adding a group of numbers and dividing the sum by the number of items added. The mean is the best known and most used measure of central tendency. The group of numbers is sometimes referred to as the data or data set. The mean measures the central location of the values within the database The mean is useful for predicting but only where there are no extreme values The Mean (or Average) 29 Arithmetic Mean (average) Create a metric for each code using the same method i.e., divide the charge by the RVU Add each of the results together to get a grand total Divide the grand total by the number of samples Pros: Easy to calculate Eliminates frequency bias Cons: Does not take into account the frequency of occurrence Not accurate if data is not normally distributed 30 xw xi w i=1 = n wi i1 Code CF Total Count Average n i 10

11 Mean Sensitivity to Outliers 31 CF Without Outliers 77.6 CF With Outliers 89.8 Median The median is the middle number in a set of data that is arranged in either ascending or descending order. One-half of the numbers will be on either side of the median. The median is good for use with non-normally distributed data as it is far less affected by outliers Order the data in ascending order Count the number of records and divide by two Pick the middle number If an even number of records, get the average of the middle two Example of Median Calculation 33 Odd Number of Records Even Number of Records Code Fee Frequency RVU CF , , The median is Code Fee Frequency RVU CF , , , The median is halfway between and ( ) / 2 =

12 Median is Less Sensitive to Outliers 34 CF Without Outliers 80.0 CF With Outliers 81.0 Mean vs. Median 35 This table shows that the median wage is substantially less than the average wage. The reason for the difference is that the distribution of workers by wage level is highly skewed. 0.01% = $23,846, % = $2,802, % = 1,019, % = 161,139 Mode The mode is the one value within a data set that is reported the most There can be more than one mode A single mode is called Unimodal Two modes is called Bimodal Many modes is called MultiModal A multimodal distribution can indicate groups of variables with difference characteristics within the same data set Paid amounts for E&M visit vs. Surgical procedures 36 12

13 Mean Median /17/2015 UniModal 37 Histogram of Procedure Code Frequency Procedure Code Bimodal 38 Summary Report for Mean Universe Anderson-Darling Normality Test A-Squared P-Value <0.005 Mean StDev Variance Skewness Kurtosis N 1014 Minimum st Quartile Median rd Quartile Maximum % Confidence Interval for Mean % Confidence Interval for Median % Confidence Interval for StDev % Confidence Intervals Multimodal 39 Histogram of Payment Frequency Payment

14 Percentiles th The p percentile is the n*p 100 Percentiles report the value for a given variable where a certain percent of observations fall above and below the value For example: At the 20 th percentile, some 20% of values are lower and some 80% of value are higher A percentile is 1/100 of the total of an ordered data set Splits the data into hundredths All data are ordered around the median (50 th percentile) th 40 observation, when the set of observations are arranged in order or magnitude; where n is the sample size. Other Standard Metrics Quartiles divide the data into four equal parts 1 st quartile = 25 th percentile 2 nd quartile = 50 th percentile (mean) 3 rd quartile = 75 th percentile Deciles divide the data into 10 equal parts 41 Quartiles and Deciles 42 Quartiles Q L i i h f i i n F, i 1, 2, 3 4 L 0 = Lower limit of the i-th Quartile class n = Total number of observations in the distribution h = Class width of the i-th Quartile class f i = Frequency of the i-th Quartile class F = Cumulative frequency of the class prior to the i-th quartile class Deciles P L i i h f i i n F, i 1, 2, 3 10 L 0 = Lower limit of the i-th Decile class n = Total number of observations in the distribution h = Class width of the i-th Decile class f i = Frequency of the i-th Decile class F = Cumulative frequency of the class prior to the i-th Decile class 14

15 Interquartile Range The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. The IQR identifies the middle 50% of the data 43 th The p percentile is the Example: Age Distribution n*p 100 th observation, when the set of observations are arranged in order or magnitude; where n is the sample size. For the age distribution, n = 121; The 75 th percentile for the age distribution is the (121 *75)/100 = ~ 91 st observation when the ages are arranged in an increasing order of magnitude. The 75 th percentile of the ages is therefore 31 years; the 25 th percentile, 50 th and 80 th percentile are the 31 st, 61 st, and 97 th observations respectively, as shown on the next slide. 25 th percentile 50 th percentile 75 th percentile 80 th percentile Age Frequency Cumulative Frequency Total 121 The 31 st observation falls in this group The 61 st observation falls in this group The 97 th observation falls in this group

16 Mean Median Anderson-Darling Normality Test A-Squared P-Value <0.005 Mean StDev Variance Skewness Kurtosis N 1632 Minimum st Quartile Median rd Quartile Maximum % Confidence Interval for Mean % Confidence Interval for Median % Confidence Interval for StDev /17/2015 Box Plot Quantitative Scale 75 th percentile IQR Referred to as whisker 75 th percentile Average/mean 50 th percentile/median 25 th percentile Referred to as whisker 25 th percentile IQR Individual box symbol IQR: Interquartile range, which is calculated by substracting the 25 th percentile of the data from 75 th percentile; consequently, it contains the middle 50% of the observations Box Plots 47 Other Ways to Graph Mean, Median and Mode 48 Summary Report for Fee Dotplot of Fee for % Confidence Intervals Fee Each symbol represents up to 8 observations. 16

17 Measures of Variability 49 Variance A measure of the average squared distance of possible values from the expected value (arithmetic average) 50 Code Fee Frequency RVU CF Difference Variance S = (4.986) , , (12.366) , (10.212) (5.070) Note that the difference (not squared) always adds up to zero n i=1 2 ( xi-x ) n - 1 Standard Deviation S = 51 n 2 ( xi-x ) i=1 n - 1 The measure of spread of values around the mean Code Fee Frequency RVU CF Difference Variance (4.986) , , (12.366) , (10.212) (5.070) The square root of the variance ( ) equals the standard deviation (29.643) For normally distributed data: 68.2% of the population values will fall between and % of population values will fall between and

18 Coefficient of Variance The coefficient of variance (CV) is calculated by dividing the standard deviation by the mean The coefficient of variation is a measure of spread that describes the amount of variability relative to the mean Because the coefficient of variation is unitless, you can use it instead of the standard deviation to compare the spread of data sets that have different units or different means. Large CV means that the data are more dispersed while a lower CV means that the data 52 CV for Two Different Codes Specialty Fee Stdev CV GS % CA % FP % GE % IM % OS % PM % Specialty Fee Stdev CV GS % CA % FP % GE % IM % OS % PM % Interquartile Range 54 The interquartile range (IQR) is the distance between the 75 th percentile and the 25 th percentile The IQR is essentially the range of the middle 50% of the data Because it uses the middle 50%, the IQR is not affected by outliers or extreme values IQR is often represented using Boxplots 18

19 Box Plots 55 Error and Confidence Intervals 56 x t (1-2 )(n -1)df S n pˆ Z (1-2 ) SE pˆ where SE = pˆ (1 - pˆ ) n What is Sample Error 57 In statistics, sampling error is the error caused by observing a sample instead of the whole population. [Bunns & Grove, 2009] The sample is never identical to the population Basically, all samples have some error when used to predict a value within the population 19

20 Causes of Sampling Error Population specific error Not understanding who or what to sample Sample frame error Occurs when the wrong sub-population is identified and/or used Selection error When data points are not selected correctly Non-response error Occurs when data are missing, variable fields are zero or other similar issues Sampling error Can occur when the wrong sample type is selected (e.g. SVRS, Cluster, Convenience) 58 Calculating the Margin of Error Margin of error rules go something like this: The larger the sample, the smaller the error The smaller the variance, the smaller the error SE can be calculated using two primary assessment types Attribute Variable Attribute 59 Variable Example of SE for Variable Assessment A sample of average charges for was taken from 50 practices in a given area Mean = $82.40 and STDev = $15.55 Assume normal distribution 68.26% of values between $66.85 and $97.95 SE = Stdev/sqrt(N), or 15.55/sqrt(50), or 15.55/7.07 = 2.2 The standard error for our estimate of the mean of $82.40 is $

21 Example for SE Calculation for Attribute Assessment In a chart review, a practice finds that $1,809 out of $5,742 was paid in error This equates to a paid error rate of 31.5% To calculate the sample error, we use this formula: Where p =.315 (1-p) = =.685 n = 5,742 (p(1-p)/n =.216 / 5742 = % The square root of % is 0.613% Plus and minus p = 30.88% to % 61 Example for SE Calculation for Attribute Assessment In a chart review, a practice finds that 6 out of 30 charts contained a medical necessity error This equates to a coding error rate of 20% To calculate the sample error, we use this formula: Where p =.2 (1-p) = =.8 n = 30 p(1-p)/n =.16 / 30 = The square root of is (7.3%) Plus and minus p = 12.7% to 27.3% 62 Common Z levels of confidence Z-score is a measure of standard distance in a normally distributed distribution Many believe that if the data set is large (n > 30), you can assume a normal or near normal distribution (NOT TRUE) Commonly used confidence levels are 90%, 95%, and 99% 21

22 Margin of Error (1/2 Interval) The margin of error is the z or t score times the standard error z and t values depend on how wide or narrow you want the margin of error to be The higher the value, the higher the margin of error Sample of 50, 95% margin of error Mean = 82.40, stdev = 15.55, SE = 2.20 Margin of error = (z or t) times SE z = 1.96*2.20 = 4.31 t = * 2.20 = What is a Confidence Interval (CI)? The purpose of a confidence interval is to validate a point estimate; it tells us how far off our estimate is likely to be A confidence interval specifies a range of values within which the unknown population parameter may lie Normal CI values are 90, 95%, 99% and 99.9% The width of the interval gives us some idea as to how uncertain we are about an estimate A very wide interval may indicate that more data should be collected before anything very definite can be inferred from the data This means when a sample is drawn there are?? chances in 100 that the sample will reflect the sampling frame at large within the sampling error 65 Using our average charge example: Interpreting the CI Mean = 82.40, SD= 15.55, SE = 2.20, ME = 4.42 (t-score) CI = /- 4.42, or 95% CI = $77.98 to $86.82 False Statement: I am 95% confident that the true average charge for this code is somewhere between $77.98 and $86.82 True Statement: In 95% of samples, the population mean will be somewhere between $77.98 and $

23 Example for SE Calculation for Attribute Assessment In a chart review, a practice finds that $1,809 out of $5,742 was paid in error This equates to a paid error rate of 31.5% To calculate the sample error, we use this formula: Where p =.315 (1-p) = =.685 n = 5,742 (p(1-p)/n =.216 / 5742 = % The square root of % is 0.613% Plus and minus p = 30.88% to % 67 Example for SE Calculation for Attribute Assessment In a chart review, a practice finds that 6 out of 30 charts contained a medical necessity error This equates to a coding error rate of 20% To calculate the sample error, we use this formula: Where p =.2 (1-p) = =.8 n = 30 p(1-p)/n =.16 / 30 = The square root of is (7.3%) Plus and minus p = 12.7% to 27.3% 68 Attribute 95% Confidence Interval 69 In attribute example 1, the SE was (.613%) The 95% half interval is * 1.96 = (1.2%) Plus and minus the p of 31.5% = 30.3% to 32.7% In attribute example 2, the SE was (7.3%) The 95% half interval is.073 * 1.96 = (14.3%) Plus and minus the p of 20% = 5.7% to 34.3% The confidence interval range has a huge impact in inferential statistics 23

24 95% Confidence Intervals To Change the Confidence Interval 71 To narrow the confidence interval Decrease the variability Lower your z/t score Increase the sample size To get a better level of confidence Decrease the variability Accept a broader CI (i.e., 80%) Increase the Sample Size x Z x Z s n s n CI Applications Physician Productivity In physician productivity studies, the 95% CI gives us a range of work RVU values. In 95% of the samples we take, the true mean for the population would fall somewhere between the lower and upper bound 72 Lower Work Mean Work Upper Work RVUs per RVUs per RVUs per Specialty FTE -National FTE -National FTE -National GP 4, , , EM 6, , , PD 4, , , CV 4, , , PY 3, , , GS 4, , , FP 4, , , OS 4, , ,

25 Mean Median Anderson-Darling Normality Test A-Squared 0.69 P-Value Mean StDev Variance Skewness Kurtosis N 40 Minimum st Quartile Median rd Quartile Maximum % Confidence Interval for Mean % Confidence Interval for Median % Confidence Interval for StDev /17/2015 CI Applications Auditing Summary Report for Overpaid 95% Confidence Intervals 73 If a hundred similar audits were performed, in 95 of them, the actual mean damage would be somewhere between $ and $ Assume the universe is 10,000 claims The difference between the mean and the lower bound of the 95% CI is $28.71 This translates to a difference in damage estimates of 107,020 rather than 135,730 (28,710) Frank D. Cohen fcohen@drsmgmt.com For More Information To Get the Toolbox Click on the Library tab (at top of page) Click on Workshop Toolboxes Select the Statistics toolbox Password is