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Chapter 15��Statistical Analysis of Quantitative Data��Dr Sanaa Abujilban, PhD

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Statistical Analysis

Descriptive statistics

Used to describe and synthesize data

Inferential statistics

Used to make inferences about the population based on sample data

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Descriptive Indexes

Parameter

A descriptor for a population (e.g., the average age of menses for Canadian females)

Statistic

A descriptor for a sample (e.g., the average age of menses for female students at McGill University)

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Phases of Data Analysis

Pre-analysis phases: coding and data entry.
Preliminary assessment and actions:
Assessing and handling missing data,
Assessing quantitative data quality,
Assessing bias,
Testing assumptions, and
Performing additional analysis such as reliability and count.

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Phases of Data Analysis:

Data and substantive analysis: the researcher must develop a substantive plan to reduce the temptation of going wrong during analysis of the data.
Interpretation of results: accuracy of data and its meaning.

5. Representation of data through graphs and tables

Now it is done through statistical packages such as SPSS.

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Statistics

Descriptive

1. Univaraite “frequency distribution”.

2. Measures of central tendency (mode, median, mean), and

measures of variability (SD, variance (the degree that observations are dispersed around the central value), range).

3. Bivaraite analysis (contingency table, correlation- such as

Pearson or Product Moment Correlation Coefficient, and

Spearman’s correlation coefficient).

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Descriptive Statistics: 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

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

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Frequency Polygon

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Shapes of Distributions

aspect of a distribution’s shape
Symmetry
Symmetric: if, when folded over, the two halves are superimposed on one another; With real data sets, distributions are rarely perfectly symmetric
Skewed (asymmetric): the peak is off center and one tail is longer than the other.

Positive skew (long tail points to the right; e.g. Personal income)
Negative skew (long tail points to the left; e.g. age at death)

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All the distributions in Figure are

symmetric.

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Shapes of Distributions (cont.)�aspect of a distribution’s shape

Peakedness (how sharp the peak is)
Modality (number of peaks)

Unimodal (1 peak)
Bimodal (2 peaks)
Multimodal (2+ peaks)

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Normal Distribution; (sometimes�called a bell-shaped curve)

Characteristics:

Symmetric
Unimodal
Not too peaked, not too flat

Important distribution in inferential statistics

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Question

Is the following statement True or False?

A bell-shaped curve is also called a normal distribution.�

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Answer

True

A normal distribution is commonly referred to as a bell-shaped curve.

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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

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Comparison of Measures of Central Tendency

Mode, useful mainly as gross descriptor, especially of nominal measures
Median, useful mainly as descriptor of typical value when distribution is skewed (e.g., household income)
Mean, most stable and widely used indicator of central tendency

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Distributions (cont.)

Category Percent

Under 35 9%

36-45 21

46-55 45

56-65 19

66+ 6

A Frequency Distribution Table

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Relationships of central tendency indexes in skewed distributions.

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Variability

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

Homogeneity—little variability (school B)
Heterogeneity—great variability (School A)

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Two distributions of different variability

School A has a wide range of scores

there are few students at either extreme

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Indexes of Variability

That express the extent to which scores deviate from one another

Range: highest value minus lowest value
the range for school A is approximately 500 (750 250), and the range for school B is approximately 300 (650 350).
Standard deviation (SD): indicates the average amount of deviation of values from the mean
the first step in calculating a standard deviation is to compute deviation scores for each subject. A deviation score (symbolized as x) is the difference between an individual score and the mean. If a person weighed 150 pounds and the sample mean were 140, then the person’s deviation score would be 10.

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Standard deviations in a normal distribution

There are about 3 SDs above and 3 SDs below the mean in a normal distribution.
normally distributed scores with a mean of 50 and an SD of 10
In a normal distribution, a fixed percentage of cases falls within certain distances from the mean. Sixty-eight percent of all cases fall within 1 SD of the mean (34% above and 34% below the mean). In this example, nearly 7 of every 10 scores fall between 40 and 60. Ninety-five percent of the scores in a normal distribution fall within 2 SDs from the mean.

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Standard deviations in a normal distribution.

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Bivariate Descriptive Statistics

Used for describing the relationship between two variables
Two common approaches:

Contingency tables (Crosstabs)
Correlation coefficients

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Contingency Table

A two-dimensional frequency distribution; frequencies of two variables are cross-tabulated
“Cells” at intersection of rows and columns display counts and percentages
Variables usually nominal or ordinal

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Correlation Coefficients

Indicate direction and magnitude of relationship between two variables
The most widely used correlation coefficient is Pearson’s r.
Pearson’s r is used when both variables are interval- or ratio-level measures.

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Question

The researcher subtracts the lowest value of data from the highest value of data to obtain which of the following?

Mode
Median
Mean
Range

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Answer

d. Range

The range is calculated by subtracting the lowest value of data from the highest value of data. The mode refers to the most frequently occurring score. The median refers to the point distribution above which and below which 50% of the cases fall. The mean is the sum of all the scores divided by the total number of scores.

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Correlation Coefficients (cont.)

Correlation coefficients can 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
A correlation coefficient of 0 indicates zero correlation;

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Correlation Coefficients (cont.)

The greater the absolute value of the coefficient, the stronger the relationship:

Ex: r = -.45 is stronger than r = +.40

With multiple variables, a correlation matrix can be displayed to show all pairs of correlations.

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Bivaraite Analysis : Correlation

A measure of bivariate descriptive statistics

Used to answer the question “To what extent are the variables related?”

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Bivaraite Analysis: Correlation

Spearman’s Rank Correlation – A method used for calculating CORRELATION A method used for calculating CORRELATION between VARIABLES, when the data does not follow the NORMAL DISTRIBUTION. This is therefore a non-parametric test.

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Bivaraite Analysis �2. Correlation

Pearson’s r - A method used for calculating CORRELATION - A method used for calculating CORRELATION between VARIABLES, when the data is believed to follow the NORMAL DISTRIBUTION. This is therefore a parametric test.

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Index of Correlation

Below .3= low correlation

.3-.7= medium correlation

Above .7= high correlation

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scatter plot

Graphically, the relationship between two variables is displayed on a scatter plot or scatter diagram.

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Figure 1 - top left - is an example of �a positive correlation.�

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Figure 2 is an example of a zero correlation.

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Figure 3 is an example of �a negative correlation.

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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.

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Describing Risk (cont.)

Some frequently used indexes:

Absolute Risk (AR): the proportion of people who experienced an undesirable outcome in each group.
Absolute Risk Reduction (ARR): represents a comparison of the two risks (exposed and unexposed groups)

ARR = AR for the unexposed group – AR for the exposed group

Odds Ratio (OR)

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The Odds Ratio (OR)

The odds = the proportion of people with an adverse outcome relative to those without it

e.g., the odds of ….

The odds ratio is computed to compare the odds of an adverse outcome for two groups being compared (e.g., men vs. women, experimentals vs. controls).

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Inferential Statistics

Used to make objective decisions about population parameters using sample data
Based on laws of probability
Uses the concept of theoretical distributions

e.g., the sampling distribution of the mean

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Sampling Distribution of the Mean

A theoretical distribution of means for an infinite (endless) number of samples drawn from the same population
Is 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.

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Statistical Inference—Two Forms

Parameter estimation
Hypothesis testing (more common among nurse researchers than among medical researchers)

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Question

Is the following statement True or False?

A correlation coefficient of -38 is stronger than a correlation coefficient of +32.

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Answer

True

For a correlation coefficient, the greater the absolute value of the coefficient, the stronger the relationship. So the absolute value of -.38 is greater than the absolute value of +.32 and thus is stronger.

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Estimation of Parameters

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)

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Confidence Intervals

CIs indicate the upper and lower confidence limits and the probability that the population value is between those limits.

For example, a 95% CI of 40–50 for a sample mean of 45 indicates there is a 95% probability that the population mean is between 40 and 50.

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Inferential Statistics �Hypothesis Testing

The most commonly used purpose of inferential statistics is hypothesis testing.
Scientific hypothesis –which the researcher believes will be the outcome of the study.
Null hypothesis – the hypothesis that actually can be tested by statistical methods – states that there is no difference between the groups or there is no actual relationship between the variables.

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Hypothesis Testing

Based on rules of negative inference: research hypotheses are supported if null hypotheses can be rejected.
Involves statistical decision-making to either:

accept the null hypothesis or
reject the null hypothesis

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Hypothesis Testing

Researchers compute a test statistic with their data (using appropriate formulas ) and then determine whether the statistic falls beyond the critical region in the relevant theoretical distribution.

Values beyond the critical region indicate that the null hypothesis is improbable (impossible), at a specified probability level.

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critical region

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Hypothesis Testing (cont.)

If the value of the test statistic indicates that the null hypothesis is improbable, then the result is statistically significant.
A nonsignificant result means that any observed difference or relationship could have happened by chance.
Statistical decisions are either correct or incorrect.

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If the value of the test statistic indicates that the null hypothesis is improbable, then the result is statistically significant

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critical region and statistical tests

critical region: The decision rule is to reject the null hypothesis if the test statistic falls at or beyond a critical region on the applicable theoretical distribution, and to accept the null hypothesis otherwise.
For every test statistic, there is a related theoretical distribution. Researchers compare the value of the computed test statistic to values in a table that specify critical limits for the applicable distribution.

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Errors in Statistical Decisions

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

Risk of error is controlled by the level of significance (alpha), e.g., α = .05 or .01.

Type II error: failure to reject a null hypothesis when it should be rejected; 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.
Larger samples = greater power

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Parametric and Nonparametric Tests

Parametric Statistics:

Use involves estimation of a parameter; assumes variables are normally distributed in the population; measurements are on interval/ratio scale

Nonparametric Statistics:

Use does not involve estimation of a parameter; measurements typically on nominal or ordinal scale; doesn’t assume normal distribution in the population

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Overview of Hypothesis-Testing Procedures

Select an appropriate test statistic.
Establish significance criterion (e.g., α = .05).
Compute test statistic with actual data.
Calculate degrees of freedom (df) (number of observation free to vary about a parameter) for the test statistic.
Obtain a critical value for the statistical test (e.g., from a table).
Compare the computed test statistic to the tabled value.
Make decision to accept or reject null hypothesis.

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Inferential Statistics

Purpose: Enable the researcher to generalize about the population based on data obtained from the sample.
“ Level of significance” which is typically .01 or .05….0.01 mean that only one case in 100 cases may yield erroneous findings.
Minimum level in nursing research is p = 0.05; this means that if the study were done 100 times, the decision to reject the null hypothesis would be wrong 5 times out of those 100 trials.

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Inferential Statistics

The most common used inferential statistics are:

t-test,
ANOVA,
Chi-square,
Mann-Whitney U,
correlation coefficient,
simple linear regression,
multiple regression,
ANCOVA,
MANOVA, and
path analysis.

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Commonly Used Bivariate Statistical Tests

t-Test
Analysis of variance (ANOVA)
Pearson’s r
Chi-squared test

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Question

Is the following statement True or False?

Parametric statistical testing usually involves measurements on the nominal scale.

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Answer

False

Parametric statistics usually involve measurements that are on the interval or ratio scale; nonparametric tests are used for nominal or ordinal data.

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t -Test

Tests the difference between two means
t-test for independent groups: between-subjects test

e.g., means for men vs. women

t-test for dependent (paired) groups: within-subjects test

e.g., means for patients before and after surgery

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Analysis of Variance (ANOVA)

Tests the difference between more than 2 means (3 or more groups)

One-way ANOVA (e.g., 3 groups): is used to test the effect of one independent variable (e.g., different interventions) on a dependent variable.
Multifactor (e.g., two-way) ANOVA: multiple independent variables
Repeated measures ANOVA (RM-ANOVA): within subjects: means at different points of time; is used when there are three or more measures of the same dependent variable for each subject.

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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

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Correlation

Pearson’s r is both a descriptive and an inferential statistic.
Tests that the relationship between two variables is not zero.

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Effect Size

Effect size is an important concept in power analysis.
Effect size indexes summarize the magnitude (size) 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.
By convention:

d ≤ .20, small effect

d = .50, moderate effect

d ≥ .80, large effect

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Multivariate Statistics

Statistical procedures for analyzing relationships among 3 or more variables
Two commonly used procedures in nursing research:

Multiple regression: is used to make predictions about phenomena.
Analysis of covariance (ANCOVA)

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Multiple Linear Regression

Used to predict a dependent variable based on two or more independent (predictor) variables
Dependent variable is continuous (interval or ratio-level data).
Predictor variables are continuous (interval or ratio) or dichotomous.

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Multiple Correlation Coefficient (R )

The correlation index for one dependent variable and 2 or more independent (predictor) variables: R
Does not have negative values ( from 0 to 1): shows strength of relationships, not direction
R² is an estimate of the proportion of variability in the dependent variable accounted for by all predictors.

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Analysis of Covariance (ANCOVA)

Extends ANOVA by removing the effect of confounding variables (covariates) before testing whether mean group differences are statistically significant
Levels of measurement of variables:

Dependent variable is continuous—ratio or interval level
Independent variable is nominal (group status)
Covariates are continuous or dichotomous

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Visual respresentation of analysis of covariance

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Question

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

Pearson’s r
Chi-squared test
t-Test
ANOVA

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Answer

b. Chi-squared test

The chi-squared test evaluates the difference in proportions in categories within a contingency table, comparing the observed frequencies with the expected frequencies. Pearson’s r tests that the relationship between two variables is not zero. The t-test evaluates the difference between two means. The ANOVA tests the difference between more than 2 means.

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Logistic Regression

Analyzes relationships between a nominal-level dependent variable and 2 or more independent variables
Yields an odds ratio—the risk of an outcome occurring given one condition, versus the risk of it occurring given a different condition
The OR is calculated after first removing (statistically controlling) the effects of confounding variables.

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Factor Analysis

Used to reduce a large set of variables into a smaller set of underlying dimensions (factors)
Used primarily in developing scales and complex instruments

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Multivariate Analysis of Variance�

The extension of ANOVA to more than one dependent variable
Abbreviated as MANOVA
Can be used with covariates: Multivariate analysis of covariance (MANCOVA)

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Causal Modeling

Tests a hypothesized multivariable causal explanation of a phenomenon
Includes:

Path analysis
Structural equations modeling

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Univariate Tests

Level of Measurement	Test
Categorical	Chi square test
Non-categorical	One-sample t test

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Bivariate Tests

VARIABLE 1 (IDV)	VARIABLE 2 (DV)	STATISTICAL TEST
Categorical	Categorical	Chi square test for crosstables
Categorical (2 groups or values)	Non-categorical	Independent samples t-test
Categorical (> 2 groups or values)	Non-categorical	ANOVA
Non-Categorical	Non-Categorical	Correlation/ Regression

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PREDICTOR VARIABLE (S)	OUTCOME VARIABLE
	Categorical	Continuous
Categorical	Chi-square, Log linear, Logistic	T-test, ANOVA (Analysis of Varirance), Linear regression
Continuous	Logistic regression	Linear regression, Pearson correlation
Mixture of Categorical and Continuous	Logistic regression	Linear regression, Analysis of Covariance

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Summary of Tests

Chi-Square : is used to look at the statistical significance of an association between a categorical outcome and a categorical determining variable.
T-Test: looks at the difference in means of a continuous variable between two groups when the determining variable is categorical .
ANOVA (Analysis of Variance) : is used to see an association between a continuous outcome variable and a categorical determining variable. Looks at the difference in means of a continuous variable between more than two groups.

- One -way ANOVA- is used for a quantitative dependent variable by a single factor (independent) variable.

- Two-way ANOVA- is used for one dependent variable by one or more factors and/or variables.

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Summary of Tests

ANCOVA (Feature of Multiple Reg. and ANOVA); use to compare means of two or more groups with a statistical control of extraneous variables (covariates).
MANOVA: An extension of ANOVA; compare 2 ore more groups that have more than one dependent variable simultaneously.
Linear Regression: estimates the coefficients of the linear equation, involving one or more independent variables, that best predict the value of the dependent variable.
Correlations: Correlation testing usually runs a continuous outcome against a continuous determining variable to see if they have a linear relationship (positive, negative or none). This type of test will not tell the strength of the relationship between the variables, but will indicate the existence of the relationship. The Bivariate Correlations procedure computes Pearson’s correlation coefficient, Spearman’s rho and Kendall’s tau-b with their significance levels.

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Guidelines for Critiquing Quantitative Analyses

1. Does the report include any descriptive statistics? Do these statistics sufficiently describe the major characteristics of the researcher’s data set?

2. Were indices of both central tendency and variability provided in the report? If not, how does the absence of this information affect the reader’s understanding of the research variables?

3. Were the correct descriptive statistics used (e.g., was a median used when a mean would have been more appropriate)?

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Guidelines for Critiquing Quantitative Analyses

4. Does the report include any inferential statistics? Was a statistical test performed for each of the hypotheses or research questions? If inferential statistics were not used, should they have been?

5. Was the selected statistical test appropriate, given the level of measurement of the variables?

6. Was a parametric test used? Does it appear that the assumptions for the use of parametric tests were met? If a nonparametric test was used, should a more powerful parametric procedure have been used instead?

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Guidelines for Critiquing Quantitative Analyses

7. Were any multivariate procedures used? If so, does it appear that the researcher chose the appropriate test? If multivariate procedures were not used, should they have been? Would the use of a multivariate procedure have improved the researcher’s ability to draw conclusions about the relationship between the dependent and independent variables?

8. In general, does the report provide a rationale for the use of the selected statistical tests? Does the report contain sufficient information for you to judge whether appropriate statistics were used?

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Guidelines for Critiquing Quantitative Analyses

9. Was there an appropriate amount of statistical information reported? Are the findings clearly and logically organized?

10. Were the results of any statistical tests significant? What do the tests tell you about the plausibility of the research hypotheses?

11. Were tables used judiciously to summarize large masses of statistical information? Are the tables clearly presented, with good titles and carefully labeled column headings? Is the information presented in the text consistent with the information presented in the tables? Is the information totally redundant?

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End of Presentation