Basic concepts of measurement

Basic Concepts in Basic Concepts in MeasurementMeasurement

R. K. PRADHANR. K. PRADHAN

Can Psychological Properties Be Can Psychological Properties Be Measured?Measured?

A common complaint: Psychological variables can’t be measured.

We regularly make judgments about who is shy and who isn’t; who is attractive and who isn’t; who is smart and who is not.

QuantitativeQuantitative

Implicit in these statements is the notion that some people are more shy, for example, than others

This kind of statement is inherently quantitative.

Quantitative: It is subject to numerical qualification.

If it can be numerically qualified, it can be measured.

MeasurementMeasurement

• The process of assigning numbers to objects in such a way that specific properties of the objects are faithfully represented by specific properties of the numbers.

• Psychological tests do not attempt to measure the total person, but only a specific set of attributes.

Measurement (cont.)Measurement (cont.)

•Measurement is used to capture some “construct”- For example, if research is needed on the construct of “depression”, it is likely that some systematic measurement tool will be needed to assess depression.

Individual DifferencesIndividual Differences• The cornerstone of psychological measurement - The cornerstone of psychological measurement - that there are real, relatively stable differences that there are real, relatively stable differences between people.between people. • This means that people differ in measurable ways This means that people differ in measurable ways in their behavior and that the differences persist in their behavior and that the differences persist over a sufficiently long time.over a sufficiently long time.

•Researchers are interested in assigning Researchers are interested in assigning individuals numbers that will reflect their individuals numbers that will reflect their differences. differences. • Psychological tests are designed to measure Psychological tests are designed to measure specificspecific attributes, not the whole person. attributes, not the whole person.

•These differences may be large or small. These differences may be large or small.

Types of Measurement Scales

1. Nominal

2. Ordinal

3. Interval

4. Ratio

Types of Measurement Scales

Nominal Scales - there must be distinct classes but these classes have no quantitative properties. Therefore, no comparison can be made in terms of one being category being higher than the other

For example - there are two classes for the variable gender -- males and females. There are no quantitative properties for this variable or these classes and, therefore, gender is a nominal variable.

Other Examples:country of originbiological sex (male or female)animal or non-animalmarried vs. single

Nominal ScaleNominal Scale

Sometimes numbers are used to designate category membership

Example: Country of Origin1 = United States 3 = Canada

2 = Mexico 4 = Other

However, in this case, it is important to keep in mind that the numbers do not have intrinsic meaning

Types of Measurement Scales

Ordinal Scales - there are distinct classes but these classes have a natural ordering or ranking. The differences can be ordered on the basis of magnitude.

For example - final position of horses in a thoroughbred race is an ordinal variable. The horses finish first, second, third, fourth, and so on. The difference between first and second is not necessarily equivalent to the difference between second and third, or between third and fourth.

Ordinal ScalesOrdinal Scales

Does not assume that the intervals between numbers are equal

Example:

finishing place in a race (first place, second place)

1 hour 2 hours 3 hours 4 hours 5 hours 6 hours 7 hours 8 hours

1st place 2nd place 3rd place 4th place

Types of Measurement Scales (cont.)

Interval Scales - it is possible to compare differences in magnitude, but importantly the zero point does not have a natural meaning. It captures the properties of nominal and ordinal scales -- used by most psychological tests.

Designates an equal-interval ordering - The distance between, for example, a 1 and a 2 is the same as the distance between a 4 and a 5

Example - celsius temperature is an interval variable. It is meaningful to say that 25 degrees celsius is 3 degrees hotter than 22 degrees celsius, and that 17 degrees celsius is the same amount hotter (3 degrees) than 14 degrees celsius. Notice, however, that 0 degrees celsius does not have a natural meaning. That is, 0 degrees celsius does not mean the absence of heat!

Types of Measurement Scales (cont.)

Ratio Scales - captures the properties of the other types of scales, but also contains a true zero, which represents the absence of the quality being measured..

For example - heart beats per minute has a very natural zero point. Zero means no heart beats. Weight (in grams) is also a ratio variable. Again, the zero value is meaningful, zero grams means the absence of weight.

Example: the number of intimate relationships a person has had

0 quite literally means nonea person who has had 4 relationships has had twice as many as someone who has had 2

Types of Measurement Scales (cont.)

• Each of these scales have different properties (i.e., difference, magnitude, equal intervals, or a true zero point) and allows for different interpretations

• The scales are listed in hierarchical order. Nominal scales have the fewest measurement properties and ratio having the most properties including the properties of all the scales beneath it on the hierarchy.

• The goal is to be able to identify the type of measurement scale, and to understand proper use and interpretation of the scale.

Evaluating Psychological TestsEvaluating Psychological TestsThe evaluation of psychological tests centers on the test’s:

ReliabilityReliability - has to do with the consistency of the instrument. A reliable test is one that yields consistent scores when a person takes the test two alternate forms of the test or when an individual takes the same test on two or more different occasions.

ValidityValidity - has to do with the ability to measure what it is supposed to measure and the extent to which it predicts outcomes.

Why Statistics?Why Statistics?

Correlation - relationship between scores

Variability - measure of the extent to which test scores differ.

Statistics are important because they give us a method for answering questions about meaning of those numbers.

Three statistical concepts are central to psychological measurement:

Prediction - forecast relationships

VariabilityVariability

• Variability is the foundation of psychological testing

• Variability refers to the spread of the scores within a distribution.

•Tests depends on variability across individuals --- if there was no variability then we could not make decisions about people.

• The greater the amount of variability there is among individuals, the more accurately we can make the distinctions among them.

VariabilityVariability

There are four major measures of variability:

1. Range - difference between the highest and lowest scoresFor Example: If the highest score was 60 & lowest was 40 = range of 20

2. Interquartile Range - difference between the 75th and 25th percentile.

3. Variance - the degree of spread within the distribution (the larger the spread, the larger the variance). It is the sum of the squared differences from the mean of each score, divided by the number of scores

4. Standard Deviation - a measure of how the average score deviates or spreads away from the mean.

Standard DeviationStandard Deviation

Standard deviation is

a measure of spread

affected by the size of each data value

a commonly calculated and used statistic

equal to iance var

typically about 2/3 of data values lie within one standard deviation of the mean.

Question: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kgFind the mean, variance and standard deviation of these weights.

Example – using individual data valuesExample – using individual data values

Answer: mean

x

Variance is the average square distance from

the mean

n

xx

6

1097664

6

42 = 7 kg

1 2 4 5 6 7 8 9 10 weight kg3

Question: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kgFind the mean, variance and standard deviation of these weights.

Example – using individual data valuesExample – using individual data values

Answer: mean n

xx

6

1097664

6

42 = 7 kg

x

Method 1 Variance n

x

22

)(

6

)710()79()77()76()76()74( 2222222 Variance is the

average square distance from

the mean

1 2 4 5 6 7 8 9 10 weight kg3

Answer: mean n

xx

6

1097664

6

42

x

Method 1 Variance n

x

22

)(

6

)710()79()77()76()76()74( 2222222

6

)3()2()0()1()1()3( 2222222

6

24

Variance is the average square distance from

the mean= 4 kg2

= 7 kg

1 2 4 5 6 7 8 9 10 weight kg3

Question: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kgFind the mean, variance and standard deviation of these weights.

Answer: mean n

xx

6

1097664

6

42

x

Method 1 Variance n

x

22

)(

6

)710()79()77()76()76()74( 2222222

6

)3()2()0()1()1()3( 2222222

6

24 = 4 kg2

standard deviation = iancevar 4

= 7 kg

1 2 4 5 6 7 8 9 10 weight kg3

Question: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kgFind the mean, variance and standard deviation of these weights.

= 2 kg

Normal Distribution CurveNormal Distribution Curve• Many human variables fall on a normal or close to normal curve including IQ, height, weight, lifespan, and shoe size.

• Theoretically, the normal curve is bell shaped with the highest point at its center. The curve is perfectly symmetrical, with no skewness (i.e., where symmetry is absent). If you fold it in half at the mean, both sides are exactly the same.

•From the center, the curve tapers on both sides approaching the X axis. However, it never touches the X axis. In theory, the distribution of the normal curve ranges from negative infinity to positive infinity.

•Because of this, we can estimate how many people will compare on specific variables. This is done by knowing the mean and standard deviation.

Scatter PlotsScatter Plots

• An easy way to examine the data given is by scatter plot. When we plot the points from the given set of data onto a rectangular coordinate system, we have a scatter plot.

• Is often employed to identify potential associations between two variables, where one may be considered to be an explanatory variable (such as years of education) and another may be considered a response variable

Relational/Correlational ResearchRelational/Correlational Research

Relational Research …

• Attempts to determine how two or more variables are related to each other.

•Is used in situations where a researcher is interested in determining whether the values of one variable increase (or decrease) as values of another variable increase. Correlation does NOT imply causation!

•For example, a researcher might be wondering whether there is a relationship between number of hours studied and exam grades. The interest is in whether exam grades increase as number of study hours increase.

Use and Meaning of Correlation Coefficients• Value can range from -1.00 to +1.00

• An r = 0.00 indicates the absence of a linear relationship.

• An r = +1.00 or an r = - 1.00 indicates a “perfect” relationship between the variables.

•A positive correlation means that high scores on one variable tend to go with high scores on the other variable, and that low scores on one variable tend to go with low scores on the other variable.

•A negative correlation means that high scores on one variable tend to go with low scores on the other variable.

•The further the value of r is away from 0 and the closer to +1 or -1, the stronger the relationship between the variables.

Coefficients of Determination

•By squaring the correlation coefficient, you get the amount of variance accounted for between the two data sets. This is called the coefficient of determination.

• A correlation of .90 would represent 81% of the variance between the two sets

of data (.90 X .90 = .81)

• A perfect correlation of 1.00 represents 100% of the variance. If you know

one variable, you can predict the other variable 100% of the time

(1.00 X 1.00 = 1.00)

•A correlation of .30 represents only 9% of the variance, strongly suggesting

that other factors are involved (.30 X .30 = .09)

Factor AnalysisFactor Analysis

Is a statistical technique used to analyze patterns of correlations among different measures.

The principal goal of factor analysis is to reduce the numbers of dimensions needed to describe data derived from a large number of data.

It is accomplished by a series of mathematical calculations, designed to extract patterns of intercorrelations among a set of variables.

Prediction/Linear RegressionPrediction/Linear Regression

• Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable, and the other is considered to be a dependent variable.

Formula : Y = a + bX ---------- Where X is the independent variable, Y is the dependent variable, a is the intercept and b is the slope of the line.

• Before attempting to fit a linear model to observed data, a modeler should first determine whether or not there is a relationship between the variables of interest

Prediction/Linear RegressionPrediction/Linear Regression• The method was first used to examine the relationship between the heights of fathers and sons. The two were related, of course, but the slope is less than 1.0. A tall father tended to have sons shorter than himself; a short father tended to have sons taller than himself. The height of sons regressed to the mean. The term "regression" is now used for many sorts of curve fitting.

• Linear regression analyzes the relationship between two variables, X and Y. For each subject (or experimental unit), you know both X and Y and you want to find the best straight line through the data.

Steps in Test Construction

1. Concept Development

2. Domain Identification

3. Item Construction

4. Item Analysis: Item Difficulty & Item Discrimination

5. Reliability: Test-Retest, Parallel Form, Slit-half, Rational Equivalence (Internal-consistency)

6. Validity: Content, Construct, Criterion

7. Norms

Comparing Reliability and Validity

Thank you.