Statistical Analysis of Quantitative Data

Purposes of statistical analysis in quantitative research

to describe the data (e.g., sample characteristics), to estimate population values, to test hypotheses, to provide evidence regarding measurement properties of quantified variables

Nominal

lowest level; involves using numbers simply to categorize attributes

Ordinal

ranks people on an attribute

Interval

ranks people on an attribute and specifies the distance between them

Ratio

highest level; ratio scales, unlike interval scales, have a meaningful zero and provide information about the absolute magnitude of the attribute

Nominal (categorical)

names; lowest level of measurement, data categorized into groups/categories, categories should be mutually exclusive

(ex. Male = 1 , Female = 2, NB = 3)

Ordinal (ranked)

2nd lowest level of measurement; numeric values on continuum but not equal, ordered and ranked but intervals not equal distance (ex: likert scale, ex: numeric pain scale 0-10-- differences cannot be specified)

Interval definition

3rd level of measurement; continuum of numeric values where intervals between numbers are equal but lacks a "true zero" (ex. fahrenheit scale 0 does not = no temperature....), datum ranked and attributed distance, between are equal, (equal intervals b/t numbers)

Ratio definition

highest level; numeric values assigned to data, begin with zero and have = intervals (ex. age, # hours studied for a test, many biophysiologic--pulse, ex. amount of money in your bank account-- 0 balance = NO money)

Levels of measurement high to low

ratio (absolute zero exists), interval (distance b/t numbers is meaningful), ordinal (numbers can be ranked or ordered), nominal (numbers are only names)

Descriptive statistics

used to describe and synthesize data; parameters and statistics

Parameters

descriptor for a population

Statistics

descriptive index from a sample

Inferential statistics

used to make inferences about the population based on sample data; assumes random sampling

Frequency distributions

a systematic arrangement of numeric values on a variable from lowest to highest and a count of the number of times (and/or percentage) each value was obtained

Frequency distributions can be described in terms of

shape, central tendency, variability

Frequency distributions presentation

in a table (Ns and percentages) or graphically (e.g., frequency polygons)

Shapes of distributions

symmetric, skewed (asymmetric)

Positive skew

long tail points to the right

Negative skew

long tail points to the left

Modality (number of peaks)

unimodal, bimodal, multimodal

Unimodal

1 peak

Bimodal

2 peaks, normal distribution (a bell-shaped curve)

Multimodal

2+ peaks

Central tendency

index of "typicalness" of a set of scores that comes from center of the distribution

Mode

the most frequently occurring score in a distribution (ex. 2, 3, 3, 3, 4, 5, 6, 7, 8, 9; mode = 3)

Median

the point in a distribution above which and below which 50% of cases fall (ex. 2, 3, 3, 3, 4 | 5, 6, 7, 8, 9; median = 4.5)

Mean

equals the sum of all scores divided by the total number of scores (ex. 2, 3, 3, 3, 4, 5, 6, 7, 8, 9; mean = 5.0)

Purpose of median

useful mainly as descriptor of typical value when distribution is skewed (e.g., household income)

Purpose of mean

most stable and widely used indicator of central tendency

Variability

the degree to which scores in a distribution are spread out or dispersed

Homogeneity

little variability

Heterogeneity

great variability

Range

highest value minus lowest value

Standard deviation (SD)

average deviation of scores in a distribution

Bivariate descriptive statistics

used for describing the relationship between two variables

Two common approaches for bivariate

crosstabs (contingency tables) and correlation coefficients

Correlation coefficients

describes intensity and direction of a relationship

Range of correlation coefficients

range from −1.00 to +1.00

Negative relationship

(0.00 to −1.00): one variable increases in value as the other decreases (e.g., amount of exercise and weight)

Positive relationship

(0.00 to +1.00): both variables increase (e.g., calorie consumption and weight)

The greater the absolute value of the coefficient...

...the stronger the relationship (ex. r - -0.45 is stronger than r = +0.40)

Correlation matrix

can be displayed to show all pairs of correlations with multiple variables

Pearson's r

(the product–moment correlation coefficient): computed with continuous measures

Spearman's rho

used for correlations between variables measured on an ordinal scale

Describing risk

clinical decision making for EBP may involve the calculation of risk indexes, so that decisions can be made about relative risks for alternative treatments or exposures

Frequently used indexes for describing risk

absolute risk, absolute risk reduction (ARR), odds ratio (OR), numbers needed to treat

Inferential statistics are used to

make objective decisions about population parameters using sample data, provide a means for drawing inferences about a population, given data from a sample, based on laws of probability, sampling error, uses the concept of theoretical distributions (ex. the sampling distribution of the mean error)

Sampling distribution of the mean

a theoretical distribution of means for an infinite number of samples drawn from the same population

Characteristics of sampling distribution of the mean

always normally distributed, its mean equals the population mean, its standard deviation is called the standard error of the mean (SEM)-- SEM is estimated from a sample SD and the sample size

Point estimation

a single descriptive statistic that estimates the population value (e.g., a mean, percentage, or OR)

Interval estimation

a range of values within which a population value probably lies-- involves computing a confidence interval (CI)

CIs reflect

how much risk of being wrong researchers take

Confidence intervals

indicate the upper and lower confidence limits and the probability that the population value is between those limits (ex. a 95% CI of 40 to 50 for a sample mean of 45 indicates there is a 95% probability that the population mean is between 40 and 50)

Hypothesis testing

helps researchers to make objective decisions about whether results are likely to reflect chance differences or hypothesized effects

Hypothesis testing involves

statistical decision making to either:

accept the null hypothesis or

reject the null hypothesis

Probability

likelihood or chance that an event will occur in a given situation; expressed as lower case p with values expressed as per cents (ex. p = 0.34)

Alpha

level of significance = probability of making a Type I error

Alpha characteristics

threshold at which statistical significance reached; determined prior to data analysis; α (alpha) = 0.05 unless otherwise stated

Decision theory

assumes all groups in a study are components of the same population; up to researcher to prove there really is a difference

Use of decision theory

to test the assumption of no difference (null hypothesis) a cut-off point is selected prior to analysis-- referred to as level of significance or alpha

Test research hypothesis

done statistically, disprove null hypothesis; reject null

Significant result

there is a difference between the two groups not as a result of chance

Non-significant result

any difference or relationship could have been purely chance

Statistically significant result

if the value of the test statistic indicates that the null hypothesis is improbable

Nonsignificant result

means that any observed difference or relationship could have happened by chance

Statistical decisions

are either correct or incorrect

Type I error

rejection of a null hypothesis when it should not be rejected; a false-positive result

Risk of error is controlled by

level of significance (alpha), e.g., α =.05 or .01

Type II error

false negative; failure to reject a null hypothesis when it should be rejected

Power

the ability of a test to detect true relationships

By convention, power should be at least

80

Larger samples =

greater power

Type I errors occur only

when findings are statistically significant (ss)

Type II errors occur only

when findings not ss

Power analysis

analysis for 4 parameters-- level of significance (alpha), sample size, power (beta), effect size; tells you how many subjects needed to start and finish the study

Hypothesis testing procedures

select an appropriate statistical test, specify the level of significance (e.g., α = .05), compute a test statistic with actual data, determine degrees of freedom (df) for the test statistic, compare the computed test statistic to a theoretical value

Bivariate statistical tests

t-tests, analysis of variance (ANOVA), chi-squared test, correlation coefficients, effect size indexes

t-Test

tests the difference between two means

t-Test for independent groups

between-subjects test

(for example, means for men vs. women)

t-Test for dependent (paired) groups

within-subjects test

(for example, means for patients before and after surgery)

Analysis of variance (ANOVA)

tests the difference between more than two means; sorts out the variability of an outcome variable into two components: variability due to the independent variable and variability due to all other sources; variation between groups is contrasted with variation within groups to yield an F ratio statistic

Types of ANOVA

one-way ANOVA (e.g., 3 groups), multifactor (e.g., two-way) ANOVA, repeated measures ANOVA (RM-ANOVA): within subjects

Chi-squared test

tests the difference in proportions in categories within a contingency table; compares observed frequencies in each cell with expected frequencies—the frequencies expected if there was no relationship

Correlation coefficient test

Pearson's r is both a descriptive and an inferential statistic; tests that the relationship between two variables is not zero

Effect size indexes

an important concept in power analysis; summarize the magnitude of the effect of the independent variable on the dependent variable; in a comparison of two group means (i.e., in a t-test situation), the effect size index is d

Effect size indexes by convention

d ≤ .20, small effect

d = .50, moderate effect

d ≥ .80, large effect

Reliability assessment

test-retest reliability, interrater reliability, internal consistency reliability

Validity assessment

content validity, construct validity, criterion validity

Research article info for hypothesis testing

the test used, the value of the calculated statistic, degrees of freedom, level of statistical significance

A researcher measures the weight of people in a study involving obesity and Type 2 diabetes. What type of measurement is being employed?

ratio

3 multiple choice options

T/F-- A bell-shaped curve is also called a normal distribution

true

The researcher subtracts the lowest value of data from the highest value of data to obtain:

range

3 multiple choice options

T/F-- A correlation coefficient of −.38 is stronger than a correlation coefficient of +.32

true

Which test would be used to compare the observed frequencies with expected frequencies within a contingency table?

chi-squared test

3 multiple choice options