Statistical Analysis and Levels of Measurement

• Purposes of statistical analysis in quantitative research are to describe data, test hypotheses, and provide evidence regarding quantified variables.

Levels of Measurement

• Nominal level involves using numbers to simply categorize attributes without quantitative meaning.

• Gender and blood type are examples of nominal measurements.

• Ordinal level ranks people on an attribute, reflecting relative standing (e.g., ability to do ADL).

• Interval level ranks people on an attribute and specifies the distance between them, IQ psychological testing is an example.

• Ratio level is the highest level, having a meaningful zero and providing information about the absolute magnitude of the attribute.

• When measuring the weight of people in a study involving obesity and type II diabetes, ratio measurement is employed because weight is a physical measure with a true zero point.

• Gender is an example of a nominally measured variable.

• Measurement of the ability to perform ADLs exemplify ordinal measurement.

• Interval measurement occurs when researchers can rank people on an attribute and specify the distance between them, like psychological testing.

Statistical Analysis

• Descriptive statistics used to describe and synthesize data.
• Parameters are descriptors for a population.
• Statistics are descriptive indexes from a sample.
• Inferential statistics involves making inferences about the population based on sample data.

Frequency Distributions

• A systematic arrangement of numeric values from lowest to highest with a count of occurrences.
• Frequency distributions are described by shape, central tendency, and variability.
• They can be presented in a table (Ns and percentages) or graphically (e.g., frequency polygons).

Shapes of Distributions

• Distributions can be symmetric or skewed(asymmetric).

• Positive skew has a long tail pointing to the right.

• Negative skew has a long tail pointing to the left.

• Modality measures the number of peaks: unimodal (1 peak), bimodal (2 peaks), multimodal (2+ peaks).

• A bell-shaped curve is also called a normal distribution and is symmetric, unimodal, and not very peaked.

Central Tendency

• Central tendency is an index of "typicalness" of a set of scores from the center of the distribution.
• Mode is the most frequently occurring score.
  • Example: in the distribution 2, 3, 3, 3, 4, 5, 6, 7, 8, 9 the mode is 3.
• Median is the point above and below which 50% of cases fall.
  • Example: in the distribution 2, 3, 3, 3, 4 | 5, 6, 7, 8, 9 the median is 4.5.
• Mean equals to the sum of all scores divided by the total number of scores.
  • Example: in the distribution 2, 3, 3, 3, 4, 5, 6, 7, 8, 9 the mean is 5.0.
• Mode is most useful as gross descriptor, especially nominal measures.
• Median is useful as descriptor of typical value when distribution is skewed (such as household income).
• Mean is the most stable and widely used indicator of central tendency.

Variability

• Variability is the degree to which scores in a distribution are spread out or dispersed.
• Homogeneity signifies little variability.
• Heterogeneity signifies great variability.
• Range is the highest value minus the lowest value.
• Standard deviation measures the average deviation of scores in a distribution.
• SD indicates the degree of error when using a mean to describe an entire sample.
• If females' anxiety SD was 7, and males' was 3, females scores were more varied, while males scores were more alike.

Bivariate Descriptive Statistics

• Used to describe relationships between two variables.

• Crosstabs (contingency tables) is a two-dimensional frequency distribution.

• Frequencies of two variables are cross-tabulated in crosstabs.

• "Cells" at the intersection of rows and columns display counts and percentages.

• Variables should be nominal or ordinal in crosstabs.

• A researcher subtracting the lowest value of data from the highest value of data will obtain the range.

Correlation Coefficients

• Correlation coefficients range from -1.00 to +1.00.
• A negative relationship (0.00 to -1.00) indicates one variable increases as the other decreases.
  • An example is the amount of exercise relating to weight.
• A positive relationship (0.00 to +1.00) means both variables increase.
  • An example is calorie consumption relating to weight.
• The greater the absolute value of coefficient, the stronger the relationship.
  • Example: r = -.45 is stronger than r = +.40.
• With multiple variables, a correlation matrix can show pairing correlations.
• Pearson’s r is a product-moment correlation coefficient, computed with continuous measures.
• Spearman’s rho is used for correlations between variables measured on an ordinal scale.
• When given the choice between correlation coefficient of -.38 vs +.32, -.38 is the stronger correlation coefficient.

Describing Risk

• Clinical decision making for EBP can involve risk index calculation for relative risks.
• Absolute risk is the proportion of people-experience undesirable outcome in each group.
• Absolute risk reduction (ARR) involves comparisons of two risks.
• Odds ratio (OR) shows the proportion of people with the adverse outcome versus those without it.
• Numbers needed to treat is the calculation of how many people would need to get the intervention to avoid one person getting the undesired outcome.
• Odds is the proportion of people with an adverse outcome relative to those without it.
• Odds ratio is used to compare the odds of an adverse outcome for two groups.
• If estimating the odds of continued smoking are 4x higher among smokers, then the OR (smoking example) for continued smoking is 4.

Inferential Statistics

• Used to make objective decisions about population parameters using sample data.
• Provide a means for inferences about a population.
• Based on law of probability using theoretical distributions.
• Sampling distribution of the mean is for example.
• Sampling Distribution of the Mean: A theoretical distribution of means for an infinite number of samples drawn from the same population.
• It is always normally distributed with the mean equal to population mean.
• Its standard deviation is called the standard error of the mean (or SEM).
• SEM is estimated from sample SD and the sample size.

Statistical Inference and Estimation of Parameters

• Point estimation is a single descriptive statistic that estimates the population value.
• Interval estimation is a range of values within which a population value probably lies.
  • Involves computing a confidence interval (CI).
  • Confidence Intervals reflect the amount of risk of being wrong that researchers take.
• Confidence intervals indicate the upper and lower confidence limits.
  • Show the probability that the population value is between those limits.
  • For example, 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

• Based on rules of negative inference: research hypotheses are supported if null hypotheses can be rejected.
• Statistical decision making accepted null hypothesis or rejected it.
• Null Hypothesis states there is no relationship between study topics.
  • An example is if there is no relationship between cardiac pain and anxiety - implying any change observed in scores is due to chance only.
• Hypothesis states there is a bidirectional relationship between study topics.
  • An example: with cardiac pain and anxiety- implying any change in score is due to a change in pain and or anxiety and not chance
• If the value of the test statistic indicates that the null hypothesis is improbable.
  • This indicates that the result is statistically significant.
• A nonsignificant result means that any observed difference or relationship could have happened by chance.
• Statistical decisions can be correct or incorrect.

Type I and Type II errors

• Type I error is the rejection of a null hypothesis when it should not be rejected.
  • It is a false-positive result
  • Risk of error is controlled by the level of significance (alpha).
  • For example α =.05 or .01.
• Type II error is a failure to reject a null hypothesis when it should be rejected.
  • It is a false-negative result.
  • The risk of this error is beta (β).
  • Power is the ability of a test to detect true relationships.
  • power = 1 - β.
  • By convention, power should be at least .80 and larger samples shows greater power.

Hypothesis Testing Procedures

• Select an appropriate test statistic;
• Establish a significance criterion:
  • For example alpha = .05 (p value);
• Compute test statistic with actual data;
• Determine degrees of freedom (or df) for the test statistic based on results.
  • Where the # of observations free to vary [n-1];
• Compare the computed test statistic to a theoretical value;
• Then accept/reject null hypothesis
  • Where Alpha (p value) greater than .05 is NS
  • Where Alpha (p) less than .05 is significant which is equal to means the results most likely not due to chance and related to intervention

Bivariate Statistical Tests

• Some Bivariate Statistical Tests are t-Tests, Analysis of variance (ANOVA), Chi-squared test, Correlation coefficients, Effect size indexes.

• t-Tests tests the difference between two means.

• t-Test for independent groups is a between-subjects test.

  • Where an example, means for the difference between men vs. women

• T-test for dependent (or paired) groups is a within-subjects test

  • An example are the means for patients before and after the procedure.

• Analysis of variance (ANOVA) tests the difference between more than two means.

• ANOVAs sorts outcome variables into components, independent variables and sources.

• Variation between groups is contrasted with variation within groups to yield an F ratio statistic.

• A Chi-Squared test can compare observed frequencies in excel to expected frequencies to ensure the presence of relationship.

• The chi-squared test would be used to compare the observed frequencies with expected frequencies within a contingency table.

Correlation Coefficients

• The value of Pearson's r provides both a descriptive statistic and an inferential application.
• Correlation Coefficients also tests that the relationship between two variables is not zero.
• Statistical significance is highly sensitive to the sample size.
  • With large samples, even very weak relationships can be statistically significant
• Effect size is a index summarizes the magnitude of independent variables effect on dependent variable.
  • Effect size can be observed in a comparison of 2 group means. If the t-test shows that the effect size index is d. By convention:
    • d ≤ .20, implies small effect
    • d = .50 implies moderate effect
    • d ≥ .80, implies large effect
• Statistical procedures for analyzing relationships among three or more variables simultaneously, commonly used in nursing research.
  • Procedures include multiple regression, analysis of covariance (ANCOVA), logistic regression, and multivariate analysis of variance (MANOVA).

Multiple Regressions

• Used to predict a dependent variable based on two or more independent (predictor) variables.
• The statistic used is the Multiple correlation coefficient, symbolized as R.
• Dependent variable is continuous (where it is interval or ratio-level data).
• Predictor variables are continuous (where it is the interval or ratio) or dichotomous.
• The correlation index is used for a dependent variable and more than two independent variables, the result is R.
• R is a non-negative numbers and signifies an index strength of direction, not direction.
• R² is an estimate of the proportion of variability in the dependent variable accounted for by all predictors.
• ANCOVA (or Analysis of Covariance) extends ANOVAs by removing the effect of confounding variables before testing.
• If the variable is Dependent it exists as continuous-ratio or interval.
• If the variable is Independent, then it exists as nominal state in groups.
• If the variable are Covariates exist, they will be continuous or dichotomous.

Logistic Regressions

• Analyzes relationships between a nominal-level dependent variable and more than two independent variables.
• Yields an odds ratio shows the risk of one condition and in comparison one for another one with its own conditions.
• Odds Ratios are calculated and adjusted to statistically control effects of confounding variables.

Reliability and Validity Assessment

• Reliability assessment shows the consistency of data.
  • Consistency is found in Test-retest reliability which shows a stability of measurability over test data.
  • Interrater reliability is measured by a test of 2 independent raters assigning the same same score to attributes after inspection.
  • Internal consistency reliability that determines if components of a scale are consistently measuring same attribute.
• Validity assesses the data's validity.
  • Valid data exist in contents.
  • Content validity shows what content in scale measure construct is.
  • Construct validity shows what measures are construct as a whole.
  • Criterion validity show the consistency of scores to a desired outcome.

Data Validity and Research

• Data validity is determined by research data analysis.
  • To assess results well, information is desired.
  • Test data show the data from the test that was used. -Value Calculations show statistical significance depending on collected data.
  • Degrees of freedom are considered on the level of level statistical significance.

Assessing Data

• Is there data and statistics.
• Were the statistics properly assessed.
• Do the statistics agree with data presented.

Critique assessment

• Do the results explain anything meaningful about what was tested.
• Were type I / II statistical errors present.
• Was analysis based on valid or unrelated data.
• Do the tables indicate an objective measure of the data.
• Statistical results by themselves do not communicate meaning.
• The meaning of the data must be explained.

Six Considerations of Interpretive Task

• Accuracy of results

• Precision of results

• The amount of effect on the important results

• Meaning of the results

• Generalizing the results

• Implication of the results

• Inferences are the results "stand-ins" for the true state of affairs.

• Statistical results do not provide the most meaningful means of communication about a study's results.

Aspects of a Research Mindset

• Evidence based data.

• Skeptic and critical based.

• Test data of "research hypothesis".

• Expect credible results.

• Data can be valid and reliable.

• A researcher supports inferences that they wish people to make, based on the research results, by ensuring study validity.

Data Precision

• Data is interpreted in light of its precision effects and sizes.
• Importance should make an effect on decisions. Statistical Hypothesis is a significant description result that has a careful analysis of data.

Data Meaning

• If the data meaning doesn't have sufficient important can have methodological issues and what it could have accomplished with the data.
• Inferences can be best show when hypothesizes have supportive results.
• Correlative tests DO NOT result in causation tests unless in a causation based design.
• Non experimental cases cannot be associated the same.
• The greatest challenges to interpreting the meaning of results comes with nonsignificant results.

Clinical significance

• Practical importance of research results in terms of genuine effects on people's lives.
• Group and clinical based significance typically make a drawing conclusion by the statistical information they have collected compared to others.
  • Effects based on (ES) are important indexes
  • There is a (Cis) measurement process as well.

The Numbers Needed To Treat (NNT)

• Shows what benchmarks show values that are high in measure.
• Clinical significance makes the benchmarks important. The measure is highly focused on the value.

Purposes of statistical analysis in quantitative research are to describe data, test hypotheses. Levels of measurement include nominal, ordinal, interval, and ratio, each providing different types of information about the measured attribute. Ratio level is the highest level, having a meaningful zero.