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Univariate Analysis

This document provides an overview of univariate analysis. It defines key terms like variables, scales of measurement, and types of univariate analysis. It describes descriptive statistics like measures of central tendency (mean, median, mode) and dispersion (range, variance, standard deviation). It also discusses inferential univariate analysis and appropriate statistical tests for different variable types and research questions, including z-tests, t-tests, and chi-square tests. Examples are provided to illustrate calculating and interpreting these statistics.

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UNIVARIATE ANALYSIS Dr. Soumya Swaroop Sahoo JR, Community Medicine PGIMS Rohtak
Contents • Introduction • Variables • Types of variables • Scales of measurement • Types of analysis • Components of univariate analysis • Advantages and limitations
Introduction • The word “statistics” has several meanings: data or numbers, the process of analyzing the data, and the description of a field of study. • It is derived from the Latin word status, meaning “manner of standing” or “position.” • Statistics were first used by tax assessors to collect information for determining assets and assessing taxes.
Introduction • Variable -any character, characteristic or quality that varies is termed a variable. • E.g. - to collect basic clinical and demographic information on patients with a particular illness. The variables of interest may include the sex, age and height of the patients.
Nominal Categories are mutually exclusive and unordered e. g- Sex (M/F) Blood group (A/B/AB/O) Ordinal Categories are mutually exclusive and ordered e.g- Disease severity (mild/moderate/ severe) Discrete Integer values, typically counts No notion of magnitude e.g- no. of children vaccinated, days sick per year VARIABLE Categorical (Qualitative) Numerical (Quantitative) Continuous Takes any value in a range of values Have a magnitude e. g- weight in kg, Height in cm
Scales of measurement • Nominal or categorical • Ordinal • Interval • Ratio
Nominal scale • Simplest level of measurement when data values fit into categories. • Observations are dichotomous or binary in that the outcome can take on only one of two values: yes or no. • Mutually exclusive. • E.g sex of patient(M/F), nationality
Ordinal scale • When an inherent order occurs among the categories, the observations are said to be measured on an ordinal scale. • Clinicians often use ordinal scales to determine a patient's amount of risk or the appropriate type of therapy. • E.g socio-economic class, rank order of a class(1st,2nd, 3rd) VAS( visual analog scale) for pain
Scales of measurement • Interval Scale • Data classified by ranking. • Quantitative classification . • Zero point of scale is arbitrary (differences are meaningful). • Fahrenheit temp. scale , Time • Ratio Scale • Data classified as the ratio of two numbers. • Quantitative classification . • Zero point of scale is absolute (data can be added, subtracted, multiplied, and divided). • E. g- Kelvin temp. scale, Weight, Height
Statistics InferentialDescriptive Collecting Organising Summarising Presenting Data Making inference Hypothesis testing Determining relationships Making predictions
Descriptive statistics • Descriptive statistics is the term given to the analysis of data that helps describe, show or summarize data in a meaningful way such that patterns might emerge from the data. • Does not allow us to make conclusions beyond the data we have analysed or reach conclusions regarding any hypotheses we might have made. • Simply a way to describe the data.
Inferential statistics • Inferential statistics is concerned with making predictions or inferences about a population from observations and analysis of a sample. • We can take the results of an analysis using a sample and can generalize it to the larger population that the sample represents.
Three types of analysis • Univariate analysis – the examination of the distribution of cases on only one variable at a time (e.g., weight of college students) • Bivariate analysis – the examination of two variables simultaneously (e.g., the relation between gender and weight of college students ) • Multivariate analysis – the examination of more than two variables simultaneously (e.g., the relationship between gender, race and weight of college students)
Purpose of diff. types of analysis • Univariate analysis – Purpose: mainly description • Bivariate analysis – Purpose: determining the empirical relationship between the two variables • Multivariate analysis – Purpose: determining the empirical relationship among multiple variables
Choosing the Statistical Technique Specific research question or hypothesis Determine # of variables in question Univariate analysis Bivariate analysis Multivariate analysis Determine level of measurement of variables Choose univariate method of analysis Choose relevant descriptive statistics Choose relevant inferential statistics
UNIVARIATE ANALYSIS INFERENTIAL STATISTICSDESCRIPTIVE STATISTICS 1)Measures of central tendency Mean Median Mode 2)Measures of dispersion Range Variance Standard deviation 1)Z Test 2)T test 3)Chi-square test
Mean • Arithmetic, or simple, mean is used most frequently in statistics. • Arithmetic average of the observations.
Median • The median is the middle observation, that is, the point at which half the observations are smaller and half are larger. • Used in place of mean when the data is skewed (not sensitive to outliers) • E.g –median price of housing in real estate, median household income
Mode • Value that occurs most frequently. • Commonly used for a large number of observations when the researcher wants to designate the value that occurs most often. • Bimodal: When a set of data has two modes • For frequency tables mode is estimated by the modal class.
Example of mean median mode related to incomes
Shapes of distribution Normal distribution: symmetrical Bell-shaped curve Positively skewed: tail on the right, cluster towards low end of the variable Negatively skewed: tail on the left, cluster towards high-end of the variable symmetrical Bimodality: A double peak asymmetrical
Position of mean median mode The mean is used for numerical data and for symmetric (not skewed) distributions. • The median is used for ordinal data or for numerical data if the distribution is skewed. • The mode is used primarily for bimodal distributions. • The geometric mean is Mean>median>modeMode>median>mean Mode>median>mean
Averages: advantages and disadvantages Types of average Advantage Disadvantage Mean 1.Uses all data values 2.Algebraically defined 1.Distorted by outliers 2.Distorted by skewness Median 1.Not distorted by outliers 2.Not distorted by skewness 1.Ignores most of the information 2.Not Algebraically defined Mode 1.Easily determined for categorical data 1.Not Algebraically defined Geometric mean 1.Appropriate for skewed data 1.Only appropriate if log transformation produces a symmetrical distribution
Range • Difference between minimum and maximum value in a data set • Larger range usually (but not always) indicates a large spread or deviation in the values of the data set. • (73, 66, 69, 67, 49, 60, 81, 71, 78, 62, 53, 87, 74, 65, 74, 50, 85, 45, 63, 100) • Range- 100-45 =55 • Range defines the normal limits of a biological characteristic. • e. g- systolic BP – 100-140 mm of Hg diastolic BP - 80-90 mm of Hg Urea - 15-40 mg
Variance • A measure of how data points differ from the mean – Measure of dispersion for a given score, The larger the variance is, the more the scores deviate, on average, away from the mean • Average of the squared differences from the mean – difficult to interpret because it is in the units of square of variables For population variance,   2 2 x X N     For sample variance,   2 2 1 x X s n    
Standard Deviation • A measure of variation that gives spread of data about the mean • Square root of the variance • higher standard deviation indicates higher spread, less consistency, and less clustering. • sample standard deviation: • population standard deviation: • measured in the same units as the data, making it easy to interpret.   2 1 x X s n       2 x N     
Relative or Standard Normal Deviate or Variate (Z) • Deviation from the mean in a normal distribution or curve • Given by symbol “Z” • Z = 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏−𝑴𝒆𝒂𝒏 𝑺𝑫 • Indicates how much an observation is bigger or smaller than mean in units of SD • 95% confidence level: Z = 1.96 • 99% confidence level: Z = 2.575 • 99.9 confidence level: Z = 3.27 •
Normal distribution curve with SD
Inferential univariate analysis components Test Description Test Statistic Compare an Observed Mean with Some Predetermined Value Z or t-test Compare an Observed Frequency with a Predetermined Value X 2 Compare an Observed Proportion with Some Predetermined Value Z or t-test for Proportions
Type of variable Nominal proportions t test of proportion NominalOrdinalInterval or ratio Kolmogorov- Smirnov test Chi square testZ test or t test
Drawing sample from population
Interval estimation Confidence interval (CI) Provide us with a range of values that we believe, with a given level of confidence, containes a true value CI for the population means n SD SEM SEMxCI SEMxCI    58.2%99 96.1%95
Testing of hypotheses learning objectives: » to understand the role of significance » to distinguish the null and alternative hypotheses » to interpret p-value, type I and II errors
Hypothesis testing • Hypotheses are defined as formal statements of explanations stated in a testable form. • To test statistical hypotheses two presumptions are made to draw the inference from sample value. • Logic- designed to detect significant differences: differences that did not occur by random chance. Formulate hypotheses Collect data to test hypotheses Accept hypothesis Reject hypothesis
Null and alternate hypothesis 1. Null Hypothesis (H0) – The difference is caused by random chance. – The H0 always states there is “no significant difference.” it means that there is no significant difference between the population mean and the sample mean. 2. Alternate hypothesis (H1) – “The difference is real”. – (H1) always contradicts the H0. • One (and only one) of these explanations must be true.
Testing Hypotheses: The Five Step Model 1. Make Assumptions and meet test requirements. 2. State the null hypothesis. 3. Select the sampling distribution and establish the critical region. 4. Compute the test statistic. 5. Make a decision and interpret results.
Two-tailed vs. One-tailed Tests • In a two-tailed test, the direction of the difference is not predicted. • A two-tailed test splits the critical region equally on both sides of the curve. • In a one-tailed test, the researcher predicts the direction (i.e. greater or less than) of the difference. • All of the critical region is placed on the side of the curve in the direction of the prediction.
The Curve for Two- vs. One-tailed Tests at α = .05: Two-tailed test: “is there a significant difference?” One-tailed tests: “is the sample mean greater than µ or Pu?” “is the sample mean less than µ or Pu?”
Two tailed test
One tailed test
Type I and Type II Errors • Type I, or alpha error: – Rejecting a true null hypothesis • Type II, or beta error: – Failing to reject a false null hypothesis
Testing of hypotheses Type I and Type II Errors. Example Suppose there is a test for a particular disease. If the disease really exists and is diagnosed early, it can be successfully treated If it is not diagnosed and not treated, the person will become severely disabled If a person is erroneously diagnosed as having the disease and treated, no physical damage is done. To which type of error we are willing to risk ?
Testing of hypotheses Type I and Type II Errors. Example. Decision No disease Disease Not diagnosed OK Type II error Diagnosed Type I error OK treated but not harmed by the treatment irreparable damage would be done Inference: to avoid Type error II, have high level of significance
Z-test • For testing the significance of differences between two means of samples • It calculates the probability that the two samples could have drawn from populations of the same mean, differences arising merely from sampling variability. • Assumptions for z-test • Population approximately Normal or large sample. Sample size must be larger than 30 • Data must be quantitative • The variable is assumed to follow normal distribution in the population
Example (z test of proportion) • In a recent survey, 55% of the population were found to be aware of modes of HIV transmission. A random sample of 150 urban persons showed that 49% of them were aware of the same. Is the difference significant? • Using the formula for proportions and 5 step method to solve…
Solution: Step 1: Random sample The sample is large (>30) Step 2: H0: Pu = .55 (converting % to proportion) ( H0: Ps = Pu ) H1: Pu≠ .55 Step 3: The sample is large, so we use Z distribution Alpha (α) = .05 Critical Z = ±1.96
Solution (cont.) Step 4 Step 5 • Z (obtained) < Z (critical) • Fail to reject Ho. There is no significant difference between the state population and the urban sample. 48.1 150/)55.1(55. 55.49. /)1(        nPP PP Z uu us
t test • When the sample size is small (approximately < 30) then the Student’s t distribution should be used • The test statistic is known as “t”. • The curve of the t distribution is flatter than that of the Z distribution but as the sample size increases, the t-curve starts to resemble the Z-curve • t Z as n increases, If n>100, t approaches Z.
Example of t test • A random sample of 26 sociology graduates scored 458 on the advanced sociology test with a standard deviation of 20. Is this significantly different from the population average (µ = 440)?
Solution (using five step model) • Step 1: Make Assumptions and Meet Test Requirements: • 1. Random sample • 2. Level of measurement is interval-ratio • 3. The sample is small (<30)
Solution (cont.) Step 2: State the null and alternate hypotheses. H0: null hypothesis µ = 440 (or H0: = μ) H1:alternate hypothesis µ ≠ 440
Solution (cont.) • Step 3: Select Sampling Distribution and Establish the Critical Region 1. Small sample, I-R scale of measurement, so t test to be used. 2. Alpha (α) = .05 3. Degrees of Freedom = n-1 = 26-1 = 25 4. Critical t = ±2.060
Solution (cont.) • Step 4: Using Formula to Compute the Test Statistic 5.4 126 20 440458 1        n S t 
Looking at the curve for the t distribution Alpha (α) = .05 t= -2.060 c t = +2.060 c t= +4.50 I
Step 5 Make a Decision and Interpret Results • The obtained t score fell in the Critical Region, so we reject the H0 { t (obtained) > t (critical) } – If the H0 were true, a sample outcome of 458 would be unlikely. – Therefore, the H0 is false and must be rejected. • Sociology graduates have a score that is significantly different from the general student body at (t = 4.5, df = 25, α = .05).
Main Considerations in Hypothesis Testing: • Sample size – Use Z for large samples, t for small (<100) • There are two other choices to be made: – One-tailed or two-tailed test • “Is there a difference?” = 2-tailed test • “Is the difference less than or greater than?” = 1-tailed test • Alpha (α) level – .05, .01, or .001? (α=.05 is most common)
Selected nonparametric tests Chi-Square goodness of fit test. To determine whether a variable has a frequency distribution comparable to the one expected expected frequency can be based on • theory • previous experience • comparison groups 2 )( 1 eioi ei ff f   
Selected nonparametric tests Chi-Square goodness of fit test. Example The average prognosis of total hip replacement in relation to pain reduction in hip joint is excellent - 80% good - 10% medium - 5% bad - 5% In our study of we had got a different outcome excellent - 95% good - 2% medium - 2% bad - 1% expected observed Do observed frequencies differ from expected ?
Selected nonparametric tests Chi-Square goodness of fit test. Example fe1= 80, fe2= 10, fe3=5, fe4= 5; fo1= 95, fo2= 2, fo3=2, fo4= 1; 2= 14.2, df=3 (4-1) 0.001 < p < 0.01 Null hypothesis is rejected 2 > 9.84 p < 0.05 2 > 11.34 p < 0.01 2 > 16.27 p < 0.001
Advantages • Simpler model – Easier to build, test and understand than other models • More reliable – More reliable as only one variable is used. • Descriptive method – A univariate model is a strong descriptive method. Analysts can change one variable each time the model is run to obtain results that show “what if” scenarios. For example, changing the variables from age to income can show different results which describe what happens when one factor changes within the model.
Disadvantages • Not comprehensive – A univariate model is less comprehensive compared to multivariate models. In the real world, there is often more than just one factor at play and a univariate model is unable to take this into account due to its inherent limitations. • Does not establish relationships – As only one variable can be changed at a time, univariate models are unable to show relationships between different factors.
References • Methods in Biostatistics by BK Mahajan • Statistical Methods by SP Gupta • Basic & Clinical Biostatistics by Dawson and Beth • Park’s textbook of preventive and social medicine 22nd edition
Univariate Analysis