Univariate and Multivariate Analysis

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Univariate and Multivariate Analysis
Suresh Rathi Program Consultant The INCLEN Trust
International New Delhi 110020 suresh_at_inclentrus
t.org
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• We owe a lot to the Indians, who taught us how
  to count, without which no worthwhile scientific
  discovery could have been made
  Albert Einstein

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STATISTICS

• Defined as the
• Collection
• Compilation
• Presentation
• Analysis
• Interpretation
• OF DATA
• When it applies to medical sciences-
  Bio-statistics

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• TYPES OF DATA

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Data Analyses

• Descriptive Statistics
• Frequency Distributions and Cross -Tabulations
• Measures of central tendency and dispersion
• Univariate / Bivariate Analysis
• t-tests and Analysis of Variance (ANOVA)
• Chi-square test
• Multivariate Analysis
• To adjust for simultaneous effects of multiple
  factors or to control the effects of confounding
  factors on the outcome variable.

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Descriptive analysis

• In the first step descriptive analysis will be
  done,
• Summarizing demographic variables by computing
  means with standard deviation for continuous
  variables and
• Percentages for categorical variables.

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Univariate analysis

• t-tests and Analysis of Variance (ANOVA)
• Chi-square test
• Univariate logistic regression analysis will be
  conducted by comparing two variables for each
  variable of interest using odds ratio (OR) and
  their 95 confidence intervals (CI).

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Epidemiology
Observation, measurement, analysis, correlation,
interpretation

• .the study of Distribution and Determinants of
  diseases

How many? In whom? Where?, When?
What, How, Why
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Definitions

• HYPOTHESIS
• A statement of belief used in the evaluation of
  population values
• NULL HYPOTHESIS (Ho)
• A claim that there is no difference b/w
  population mean (?) hypothesized value (?o)
• ALTERNATE HYPOTHESIS (H1)
• A claim that disagrees with the Null Hypothesis
• TEST STATISTIC
• A statistic used to determine the relative
  position of the mean in the hypothesized
  probability distribution of sample means.

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Definitions

• CRITICAL REGION (REJECTION REGION)
• Region on the far end of the distribution
• If only one end of the distribution is involved,
  the region is referred to as one-tailed test.
• If both ends are involved, the region is known as
  two-tailed test.
• When the computed value falls in the critical
  region, we reject the null Hypothesis.
• The probability that a test statistic falls in
  the critical region is denoted by ?
• SIGNIFICANCE LEVEL
• Level that corresponds to the area in critical
  region.
• When a test statistics falls in this area the
  result is called as Significant at ? level

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Definitions

• P-VALUE
• Area in the tail(s) of a distribution beyond the
  value of the test statistic.
• The probability that value of calculated test
  statistic or a more extreme one, occurred by
  chance alone is denoted by p
• NON-REJECTION REGION
• Region of the sampling distribution not included
  in ?. That is located under the middle portion
  of the curve.
• Non-Rejection Region is denoted by (1- ? )
• TEST OF SIGNIFICANCE (Hypothesis Test)
• Procedure used to establish the validity of a
  claim by determining whether or not the test
  statistic falls in the critical region. If it
  does, the results are referred to as Significant.

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PROCEDURE FOR TEST OF SIGNIFICANCE (STEPS)

• I. State Null versus Alternate Hypothesis
• Ho ? ?o
• H1 ? ?o
• H1 ? ? ?o, H1 ? ?o
• II. Choose a significance Level
• ? ?o (?o 0.05 or 0.01)
• III. Compute the test Statistic (Z-test, t-test)
  
• x ?
• Z --------------
• ? / n
• x ?
• t --------------
• s / n

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PROCEDURE FOR TEST OF SIGNIFICANCE (STEPS)

• IV. Determine the critical Region
• Which is the region of Z-distribution or
  t-distribution with ?/2 in each tail.
• V. Reject the null Hypothesis if the test
  statistic falls in the rejection Region
• Do not reject the null Hypothesis if it falls in
  the non-rejection Region
• VI. State appropriate conclusion

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t- Distribution

• Unimodal
• Bell Shaped
• Symmetrical
• Extends initially in either direction
• An area under curve is equal to 1.0 (100)
• Areas under curve (?) and are a function of
  quantity called degrees of Freedom (df)
• df n-1
• df Measures the quantity of information
  available in ones data that can be used in
  estimating the Population Variance (?2).
• Uses
• When population SD is not known
• Sample size less than 25

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EXAMPLE

• A smog alert is issued when the amount of
  particular pollutant in the air is found to be
  greater than 7ppm. Samples collected from 16
  stations given an X of 7.84 with an S of 2.01. Do
  these findings indicate that the smog alert
  criterion has been exceeded or can the results be
  explained by chance?

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• 1. Ho ? ? 7.0 H1 ? gt 7.0
• 2. ? 0.05
• 3. Test Statistic
• X - ? 7.84 - 7
• t ---------------- ------------------
  1.68
• s/ n 2.01/ 16
• 4. Critical Region
• Since H1 gt 7.0 indicates one tailed tests. We
  place all of ? 0.05 on the VE side.
• From table of t distribution we find that,
• Df 15
• t 1.753

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• 5. Since calculated t 1.68 does not fall in
  critical region we do not reject Ho.
  Alternatively, we conclude the data were
  insufficient to indicate that the critical air
  pollution level of 7ppm.

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Chi-Square Test (X2)

• For Qualitative Data
• Smoker or Non-Smoker
• Normotensive or Hypertensive
• ? ( O E )2
• X2 ----------------
• E
• df Degree of Freedom
• (c-1) (r-1) (Columun 1) (Row 1)

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For Example

• In a study we find that 76 out of 100 children
  treated with Vit C and 63 of 100-placebo group
  caught cold. Does the developing cold differ
  b/w the two groups.

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• 1. Ho The two groups are homogeneous in their
  cold developing pattern.
• H1 The two groups are not homogeneous in their
  cold developing pattern.
• 2. ? 0.05
• 3. Critical Region?
• X2 (c-1) (r-1)

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• 4. Test Statistic
• ? ( O E )2
• X2 ----------------
• E
• Row Total Column Total
• Expected Value ---------------------------------
  ----
• Grand Total

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• O E O-E (O-E)2 (O-E)2
• 
  --------------------------------------------------
  ---------------------------------
• 76 69.5 - 6.5 42.25 0.608
• 63 69.5 - 6.5 42.25 0.608
• 24 30.5 6.5 42.25 1.385
• 37 30.5 6.5 42.25 1.385
• ?3.986

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Multivariate analysis

• Multiple models
• Linear regression
• Logistic regression
• Cox model
• Poisson regression
• Loglinear model
• Discriminant analysis
• Choice of the tool according to the objectives,
  the study, and the variables

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Multiple Regression
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Multiple Regression
Regression Analysis is the
estimation of the linear relationship between a
dependent variable and one or more independent
variables or covariates.
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Multiple Regression

• Linear
• Logistic
• Independent variables
• Dependent variable

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Simple linear regression

• Relation between 2 continuous variables (SBP and
  age)
• Regression coefficient b1
• Measures association between y and x
• Amount by which y changes on average when x
  changes by one unit
• Least squares method

Slope
y
x
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Multiple linear regression

• Relation between a continuous variable and a set
  of i continuous variables
• Partial regression coefficients bi
• Amount by which y changes on average when xi
  changes by one unit and all the other xis
  remain constant
• Measures association between xi and y adjusted
  for all other xi
• Example
• SBP versus age, weight, height, etc

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Multiple linear regression

• Predicted Predictor variables
• Response variable Explanatory variables
• Outcome variable Covariables
• Dependent Independent variables

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• Multiple Logistic Regression

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Multivariate analysis

• Before conducting multivariate analysis,
  association among independent variables will be
  checked by chi-square test. All the variables
  meeting the selection criteria will be entered
  one by one, starting with the highly significant
  factor from the univariate analysis.
• Selection of final model will be based on
• Parsimony, (good sense)
• Biological interpretability and
• Statistical significance.
• The adjusted odds ratios (ORs) and their 95
  confidence intervals (CIs) will be computed using
  the estimates of parameters of final model. The
  dependent variable will be dichotomous,
• P-values will be noted to assess the model fit.

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THANKS