Statistics

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Branch of applied mathematics that deals with collecting, organizing & interpreting data using well-defined procedures in order to make decisions....

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Statistics

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an estimate of the population, calculated from data in a sample drawn from the population...

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Statistic

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the term used to describe a characteristic of a population...

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parameter

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____ data types can only take on certain values.

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Discrete data types.
ex:
-Nominal (Categorical)
-Ordinal

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____ data types can take on (almost) any value; often reported using specific values (ie. BP)

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Continuous:
ex:
-Interval
-Ratio

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Nominal (categorical) Data =

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non-orderable qualitative discrete categories

ex: race, ethnicity, gender

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Ordinal Data =

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reflects a "greater or lesser" degree of something, or may reflect a "precedes" or "superior" concept.

-numbers or discrete categories can be placed in a meaningful order
-interval between categories cannot be assumed to be equal.

ex: SES 1= low, 2= medium 3= high

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2 types of Ordinal Data

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Ordered-categories Data
-ex: cancer stages
Rank-ordered Data
-ex: committee proposals 1st, 2nd, 3rd, in order of proposal, not importance.

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2 types of Continuous Data

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Interval Data
-data ordered in a logical sequence.
-EQUAL intervals
-isnt necessarily an absolute zero point. (e.g. IQ scores)

Ratio Data
- # continuous with equal intervals between them
-has a meaningful zero point. e.g. BP, income.

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4 common types of Descriptive Stats

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1) frequency distribution tables/graphs
2) central tendencies (mean, median, mode)
3) variability (range, SD)
4) Z-scores

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Advantages and disadvantages of Frequency Distributions

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Advantages:
presents entire set of scores

Disadvantages:
can be complex with large sets of data

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3 Descriptors of Central Tendency

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Mode
Median
Mean

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Variability definition

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The degree to which scores in a distribution are spread out or dispersed.
-Homogeneity= little variability
-Heterogeneity= great variabilty

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Deviation Score =

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How far any element in a distribution is from the mean

i.e. mean of 16, an element with a value of 23 has a deviation score of 7

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Variance (mean square) =

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mean of the squares of all the deviation scores (eliminates “-” sign)

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Standard Deviation =

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Square root of variance.
- % of elements in a normal distribution is constant for a given # of SDs above or below the mean

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Purpose of Z scores

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to identify the precise location of a specific value within a distribution by using a single number
-Indicates how many standard deviations there are between the score and the mean; can use this to compare scores from different ways of measuring the same thing (i.e. SAT vs. ACT scores

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Z scores=

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# of SDs away from the mean

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Used to make inferences about the population based on sample data

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Inferential Stats

*Done by "Stat-hypothesis-testing"

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Standard Error of the Mean (SEM) =

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SD of the "Sampling Distribution of the Means".

ex: % women in SBU PA program compared to % women in ALL PA programs

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Confidence Interval

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used to state how "confident" we are that the mean for the SAMPLE we have chosen is close to the mean for the WHOLE population.
- % of values under each part of the normal curve. (within 2 SD = 95% CI)

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Z scores vs. T scores

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Z- uses data from a single sample to test a hypothesis about the population mean in situations where the population SD is KNOWN (not the usual)

T- uses data from sample(s) to test a hypothesis about a population in situations where the population SD is UNKNOWN (we have to use an estimated SEM)

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ANOVA

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analysis of variance:
Used when more than one comparison is being made
Ex - means of more than two groups are compared: placebo and two different drugs.

Measured by F ratio=
Varianc btwn groups/variance within groups.
-type of parametric test

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3 Examples of Non-parametric tests

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1) Chi square
2) mann-whitney U test
3) Wilcoxon T test

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Used when frequency data classifies individuals with a given outcome. Differences between expected and observed.

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Chi Square

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Formula for Chi Square=

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= sum of all (F observed- F expected)^2 / F expected

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In Chi Squares, the larger the value, the more likely the value is...

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NOT due to chance.

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Mann Whiteny U test

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uses ordinal data (rank orders) from two separate samples to test a hypothesis about the difference between 2 populations or 2 treatment conditions

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Wilcoxon T test

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uses the data from a repeated measures or matched-samples design to evaluate the difference between 2 tx conditions (used as an alternative to the related-sample t test in situations where the data can be rank ordered but do not satisfy the more stringent t test requirements)

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Correlation

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used to establish and quantify the STRENGTH and DIRECTION of relationship between two variables

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Regression

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used to express the FUNCTIONAL relationship between two variables

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Pearson Correlation

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measures the degree of linear relation between 2 variables. (+ or -) indicates the direction. The magnitude (0 to 1) indicates the degree to which the data points fit on a straight line (interval or ratio data)

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2 applications of th Pearson Correlation

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Point-Biserial Correlation: used when one variable is dichotomous (i.e., has only 2 values; e.g., yes/no) and the second variable is measured on an interval or ratio scale

Phi-Coefficient: both variables are dichotomous.

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Spearman Correlation

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measures the degree to which the relationship between two variables is one-directional. Used when both variables (X and Y) are ranks measured on an ordinal scale

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Null Hypothesis definition

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(HO), usually states that there is NO difference between two (or more) populations or that two (or more) variables will NOT be correlated

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Type 1 error

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rejection of a null hypothesis when it should not be rejected—a false positive study result
risk is controlled by the level of significance (alpha), usually set at .05 or .01 (likelihood a result has occurred by chance)


*IMPORTANT

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P value

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probability of making a Type I error if Ho is rejected.

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Type II error

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failure to reject a null hypothesis when it should be rejected—a false negative study result (results were, in fact, NOT due to chance, but were related to the study variable)
Defined as β, and is usually set at 0.20

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Power

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ability of a study to detect an outcome of interest; the probability that a false null hypothesis will actually be rejected
-Power is defined as (1-β), and is usually set at .80 (80%)
-should ALWAYS be done before a study
-Tells you how large a sample you need to detect a specific amount of change (difference)

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T or F: The Null hypothesis is either rejected or accepted.

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FALSE. The null hypothesis is either rejected or NOT rejected.
It is NOT accepted.

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To get more Power you must...

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increase the sample size, collect higher-level data and/or do a paired design.

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______ statistics involves variables measured on a nominal or ordinal scale.

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Non Parametric Stats

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proportion of patients that HAVE disease (same as study pre-test probability)

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Prevalence

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proportion of patients WITH the condition who test positive for the condition

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Sensitivity

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Specificity

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proportion of patients WITHOUT the condition whose test results are NEGATIVE for the condition.

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Positive Predictive Value (PPV)

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proportion of patients with a POSITIVE test who HAVE the condition.

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Negative Predictive Value (NPV)

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proportion of patients with a NEGATIVE test who DO NOT have the condition

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Likelihood ratio:

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proportion of patients WITH disease who have a particular finding divided by the proportion of patients WITHOUT disease who have the same finding

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Shows how much the odds of disease are increased if the test result is positive

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Positive likelihood ratio
LR+

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Negative likelihood ratio LR-

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shows how much the odds of disease are decreased if the test result is negative.

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Prevalence=

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all patients with disease/
total population

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Sensitivity=

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true +/ all diseased

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Specificity=

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true - / all without disease

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PPV=

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true + / all +

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NPV=

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true - / all -

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LR+ =

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Sensitivity/ (1-specificity)

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LR- =

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(1-sensitivity)/ specificity