# LEARNING OBJECTIVES SCALES OF MEASUREMENT: A REVIEW SCALES OF MEASUREMENT: A REVIEW DESCRIBING RESULTS DESCRIBING RESULTS 8/14/2016

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1 UNDERSTANDING RESEARCH RESULTS: DESCRIPTION AND CORRELATION LEARNING OBJECTIVES Contrast three ways of describing results: Comparing group percentages Correlating scores Comparing group means Describe a frequency distribution, including the various ways to display a frequency distribution Describe the measures of central tendency and variability Define a correlation coefficient Define effect size Describe the use of a regression equation and a multiple correlation to predict behavior Discuss how a partial correlation addresses the third-variable problem Summarize the purpose of structural equation models SCALES OF MEASUREMENT: A REVIEW Whenever a variable is studied, the researcher must create an operational definition of the variable and determine what type of scale will be used to analyze the variable. Do you want to measure your variables using a yes/no scale, will it be an overall score acquired through scale rating questions on a survey measuring a psychological construct, or will it be a physiological measurement? The scales of the variable can be described using one of four scales of measurement: nominal, ordinal, interval, and ratio. The scale used determines the types of statistics that are appropriate when the results of a study are analyzed. SCALES OF MEASUREMENT: A REVIEW Nominal No numerical, quantitative properties Levels represent different categories or groups Most independent variables in experiments are nominal. Ordinal exhibit minimal quantitative distinctions Rank the levels from lowest to highest Interval Intervals between levels are equal in size Can be summarized using mean or arithmetic average No absolute zero Ratio The variables have both equal intervals and an absolute zero point that indicates the absence of the variable being measured. Can be summarized using mean or arithmetic average Time, weight, length, and other physical measures are the best examples of ratio scales. Interval & Ration Variables are treated the same in SPSS and are called Scale variables DESCRIBING RESULTS Depending on the way that the variables are studied, there are three basic ways of describing the results: (1) comparing group percentages, (2) correlating scores of individuals on two variables, and (3) comparing group means. Comparing group percentage Here the focus is on percentages because the variable is nominal: for example, liking and disliking are simply two different categories. After describing the data, the next step would be to perform a statistical analysis to determine whether there is a statistically significant difference between two groups (for example, gender). DESCRIBING RESULTS Correlating individual scores A second type of analysis is needed when one do not have distinct groups of subjects. Instead, individuals are measured on two variables, and each variable has a range (interval/ratio) of numerical values. Comparing group means Much research is designed to compare the mean responses of participants in two or more groups. Therefore, the Independent variable is the study s groups (nominal/ordinal) which are measured for differences in the Dependent variable (interval/ratio) 1

2 FREQUENCY DISTRIBUTIONS A frequency distribution indicates the frequency or count of the number of occurrences of values for a particular variable. Graphing frequency distributions can be displayed using charts or graphs such as: Pie charts Bar graphs Histograms Frequency polygons PIE CHART Pie charts divide a whole circle, or pie, into slices that represent relative percentages. BAR GRAPH DISPLAYING DATA OBTAINED IN TWO GROUPS Bar graphs use a separate and distinct bar for each piece of information. They are used when the values on the x axis (independent variable) are nominal (categorical) variables HISTOGRAM SHOWING FREQUENCY OF RESPONSES Histograms use bars to display a frequency distribution for a quantitative variable. In this case, the scale values are continuous. Histogram vs. Bar Chart Histograms have no gaps to indicate continuous or ordinal data. Bar Graphs have gaps do indicate separate categories for nominal or categorical data. Histogram: No Gaps Bar Chart: Gaps FREQUENCY POLYGONS VS LINE GRAPHS Frequency polygons use a line(s) to represent the distribution of frequencies of scores. Line graphs - Used when the values on the x axis (independent variable) are ratio/interval (continuous) numbers Frequency Polygon Line Graph 2

3 SCORES ON AGGRESSION MEASURE IN MODELING AND AGGRESSION When reporting results, the individual score counts along with Standard Deviations, Sum of Squares, Means, and total numbers are often given. DESCRIPTIVE STATISTICS A Central tendency statistic tells us what the sample as a whole, or on the average, is like. There are three measures of central tendency the mean, the median, and the mode. Mean (M) Obtained by adding all the scores and dividing by the number of scores Indicates central tendency when scores are measured on an interval or ratio scales (continuous variables) Median (In scientific reports, the median is abbreviated as Mdn) Score that divides the group in half Indicates central tendency when scores are measured on an ordinal, interval, and ratio scales (categorical and continuous variables) Mode Most frequent score The only measure of central tendency that is appropriate if a nominal scale is used (categorical variables) The mode does not use the actual values on the scale, but simply indicates the most frequently occurring value. DESCRIPTIVE STATISTICS Variability is the amount of spread in the distribution of scores. Variance (s²) The standard deviation is derived by first calculating the variance, symbolized as s 2 The standard deviation is the square root of the variance DESCRIPTIVE STATISTICS Standard deviation (s) Abbreviated as SD in scientific reports Indicates the average deviation of scores from the mean. Range Another measure of variability is the range, which is simply the difference between the highest score and the lowest score. CORRELATION COEFFICIENTS: STRENGTH OF RELATIONSHIPS Correlation coefficient: Describes how strongly variables are related to one another. Probably most people are familiar with the Pearson product-moment correlation coefficient. CORRELATION COEFFICIENTS: STRENGTH OF RELATIONSHIPS Pearson product-moment correlation coefficient (Pearson r): Used when both variables have interval or ratio scale properties Values of a Pearson r can range from 0.00 to ±1.00 Provides information about the strength and the direction of relationship A correlation of 0.00 indicates that there is no relationship between the variables. The nearer a correlation is to 1.00 (plus or minus), the stronger the relationship. The relationship between variables can be described visually using scatterplots. Scatterplots can often show at a glance whether there is a relationship between variables. 3

4 SCATTERPLOTS OF PERFECT (±1.00) RELATIONSHIPS An upward trending line indicates a positive relationship, and a downward trending line indicates a negative relationship. SCATTERPLOT PATTERNS OF CORRELATION Positive Relationship Negative Relationship No Relationship Curvilinear Curvilinear Strong vs Weak Linear Relationships IMPORTANT CONSIDERATIONS Some important things to consider: Restriction of range occurs when the individuals in your sample are very similar on the variable you are studying. If one is studying age as a variable, for instance, testing only 6- and 7-year-olds will reduce one s chances of finding age effects. if you reduce the range of values of the variables in your analysis than you restrict your ability to detect relationships within the wider population. Full Range: Restriction of Range: IMPORTANT CONSIDERATIONS Important things to consider: Curvilinear relationship The Pearson product-moment correlation coefficient (r) is designed to detect only linear relationships. If the relationship is curvilinear, the correlation coefficient will not indicate the existence of a relationship. EFFECT SIZE Effect Size: Refers to the strength of association between variables Pearson r correlation coefficient is one indicator of effect size It indicates the strength of the linear association between two variables Advantage of reporting effect size - Provides a scale of values that is consistent across all types of studies 4

5 EFFECT SIZE These reflect the differences in effect sizes for correlations: Differences in effect sizes Small effects near r =.15 Medium effects near r =.30 Large effects above r =.40 Squared value of the coefficient (r²) transforms the value of r to a percentage When you report effect size for correlations, you report r 2 as a percentage of variance that your two variables share Hence, r² - is the Percent of shared variance between the two variables REGRESSION EQUATIONS Calculations used to predict a person s score on one variable when that person s score on another variable is already known Y = a + bx Y = Score one wishes to predict (DV) X = Score that is known (IV) a = Constant (y-intercept) b = Weighing adjustment factor (Slope) To predict criterion variable (X) on the basis of predictor variable (Y), demonstrate that there is a reasonably high correlation between the two FYI: The constant term in linear regression analysis seems to be such a simple thing. Also known as the y intercept, it is simply the value at which the fitted line crosses the y-axis. While the concept is simple, there s a lot of confusion about interpreting the constant. That s not surprising because the value of the constant term is almost always meaningless! Paradoxically, while the value is generally meaningless, it is crucial to include the constant term in most regression models! Regression Equation vs Linear Equation Regression Equation Does this formula look familiar? Y = a + bx Y = Score one wishes to predict X = Score that is known a = Constant (y-intercept) b = Weighing adjustment factor (rise/run) To predict criterion variable (X; DV) on the basis of predictor variable (Y; IV), one must demonstrate that there is a reasonably high correlation between the two Linear Equation from Algebra Class: Slope Intercept Form: Y = mx + b m = slope (rise/run) b = y-intercept MULTIPLE CORRELATION A technique called multiple correlation is used to combine a number of predictor variables to increase the accuracy of prediction of a given criterion or outcome variable A multiple correlation (symbolized as R to distinguish it from the simple r) is the correlation between a combined set of predictor variables and a single criterion variable. Symbolized as R Y = a+b 1 X 1 +b 2 X b n X n Y = Criterion variable (DV) X 1 to X n = Predictor variables (IV) a = Constant (y-intercept) b 1 to b n = Weights multiplied by scores on the predictor variables (slope) PARTIAL CORRELATION AND THE THIRD-VARIABLE PROBLEM Third-variable problem - An uncontrolled third variable may be responsible for the relationship between two variables of interest Partial correlation: Provides a way of statistically controlling for third variables A partial correlation is a correlation between the two variables of interest, with the influence of the third variable removed from, or partialed out of, the original correlation. This provides an indication of what the correlation between the primary variables would be if the third variable were held constant. This is not the same as actually keeping the variable constant, but it is a useful approximation. The outcome depends on the magnitude of the correlations between the third variable and the two variables of primary interest PARTIAL CORRELATION AND THE THIRD-VARIABLE PROBLEM Hierarchical Regression: Deals with the third-variable problem by statistically controlling for the effects of a third variable (a) and only looking at the effects of the variable of interest on the dependent variable (b) 5

6 Structural equation modeling (SEM) is a general term to refer to these statistical techniques. The methods of SEM are beyond the scope of this class, but one will likely encounter some research findings that use SEM. Therefore, it is worthwhile to provide an overview. SEM Describes expected pattern of relationships among quantitative non-experimental variables After data have been collected, statistical methods describe how well the data fits the model Structural equation modeling (SEM) uses statistical techniques known as mediation and moderation using regression techniques to identify the direction and influence that two or more variables have on another variable A moderator is a variable that affects the direction and/or strength of relationship between an independent/predictor variable and a dependent/criterion variable A mediator to the extent that it accounts for the relationship between the predictor and the criterion Mediators explain how external physical events take on internal psychological significance. Whereas moderator variables specify when certain effects will hold, mediators speak to how or why such effects occur Moderation (a.k.a. Interaction) Stress Indirect Effect Mediation Social Support Indirect Effect Social Support Psychological Adjustment Stress (Revised) Direct Effect Psychological Adjustment Path diagrams Visual representation of the model being tested Show theoretical causal paths among the variable Used to study modeling Arrows lead from variable to variable Statistics provide path coefficients Similar to standardized weights in regression equations Indicate the strength of relationship between variables in the path STRUCTURAL MODEL BASED ON DATA FROM HUCHTING, LAC, AND LABRIE Here s an example of a structural equation model. The direction of the arrows indicate the direction of a variable s influence upon another variable Notice the values along arrow pathways (recall +/- 1 indicates a perfect relationship) Resulting Structural Equation Models (SEM) can become quite intricate: LAB Go to Labs on the website and complete the following: Description & Correlation Describing Correlation Coefficients Work on Research Projects and Final Research Papers (Due December 7 th ) 6

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