Transcript Correlation

Research Designs
Collecting Information
Quantitative Research
Descriptive Research
Quantitative Method
Statistical Method
Comparative Research
Statistical Comparisons
Experimental Research
Analysis of Statistical Data
Qualitative Reseach
Characteristics
Descriptive Research: Quantitative Method
• Descriptive Analysis
– Limits generalization to the particular group of
individuals observed.
– No conclusions are extended beyond this group
– Any similarities to those outside the group can not be
assumed
– The data describe one group and that group only
– Provides information about the nature of a particular
group of individuals.
Descriptive Research: Quantitative Method
• Raw Data
– Frequencies
• Measures of Central Tendency
– Mean
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•
•
•
Used with Interval and Ratio Scaled data
Is the most stable measure of central tendency
Extreme scores may have undo influence over the results
used when the distribution of scores has approximately the same
number of extremely high and low scores
– Median
• Preferred when an Ordinal Scale is used
• Used when data is anticipated to be missing and a reporting of mean
would be misleading (slow learners example)
– Mode
• Quickest method
• Best used only when Nominal Data are being described
• Mode can fluctuate wildly with a small change in a few scores
Descriptive Research: Statistical Method
• Measures of Variability
– Range
• The difference between the values of the largest and smallest
scores in a distribution
• Changes of only a few scores can effect it greatly: least stable
measure
• Can not be used with Nominal Data
• Use should be restricted to Ordinal or Interval Data
– Standard Deviation
• Most widely used measure of Variability
– The larger the standard deviation the greater the degree of
variability
– Can never be less than 0
• The absolute value of the standard deviation has little meaning.
• SD should be considered a relative measure (i.e., SD’s of 5 and
60 mean exactly the same thing when foot and inch scales are
used.
Descriptive Research: Statistical Method
• Standard Scores
– Z-Score Scale
• Used to convert measurements from any arbitrary unit to a
common standard
• Used to allow direct comparison of individual scores obtained
from scales of measurement with quantitatively or even
qualitatively different units of measurement.
• The unit of measurement on the z-score scale is the standard
deviation of the distribution of the original measurements
• The mean of the z-score scale is 0.
– Other Standard Scores in Common Use
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•
•
•
T score: M=50, SD=10
GRE, SAT: M=500, SD=100
IQ: M=100, SD=15
ACT: M= 20, SD=5
Statistical Method: Foundations
– Type I and Type II Error
You decide he
is cheating
You decide he
is not cheating
He is cheating
He is not cheating
You are
correct
You are wrong
(Type I error –
alpha)
You are wrong
(Type II error –
beta)
You are correct
Statistical Method: Foundations
• Probability
– Definition of Probability
• The probability of an event is the number of favorable events divided
by the number of possible events
– Additive Law of Probability
• When the events are related by the word or
• Probability of pulling an Ace or King
P(ace)= 4/52 (.08) P(king)= 4/52 (.08): P(ace or king) = 8/52 (.16)
– Multiplicative Law of Probability
• When the events are related by the word and
• Probability of pulling an Ace and a King
P(ace or king)=8/52 (.16) P(what was not drawn 1st)= 4/51 (.08):
P(ace and king) = .16 x .08 = .01
Statistical Method: Foundations
• Subject Selection
– Random
• All elements in a population have an equal chance of being
chosen to participate in the sample
– Independent
• The probability of an event does not depend on previous
events (the gambler’s fallacy)
Descriptive Research: Statistical Method
• Chi Square (χ2)
– Typically used with Nominal data to compare
proportions
– Used to ascertain a probability that proportions of
representation within categories are similar or different
across groups
– Compares what was Observed to what was Expected
χ2 = Σ ((fo – fe)2 / fe)
– Assigns a probability that what was observed was not a
function of random error
Descriptive Research: Statistical Method
• Chi Square (χ2)
SPED 6370
Male
Female
SPED Grad
3
33
13
180
χ2 = Σ ((fo – fe)2 / fe)
6370
SPED
M
3
33
F
13
180
16
Compute Expected
36
193
(16 x 33) / 229 = 2.31
(16 x 193) / 229 = 13.49
(213 x 33) / 229 = 30.69
(213 x 193) / 229 = 179.52
229
213
6370
SPED
M
3 (2.31)
42 (30.69)
F
13 (13.49)
180 (179.52)
χ2 = .21 + .02 + .17 + .00 =
Compute χ2
(3 – 2.31)2 / 2.31 = .21
(13– 13.49)2 / 13.49 = .02
(42 – 30.69)2 / 30.69 = .17
(180 – 179.52)2 / 179.52 = .00
Table Lookup (1 df)
.40 (ns)
* .05 must be > 3.84
** .01 must be > 6.64
Descriptive Research: Statistical Method
• Correlation
– A measure of the relationship between two or more
paired variables or two or more sets of data
– The degree of relationship may be measured and
represented by the coefficient of correlation: -1 to +1
– Represented by either the letter r or the Greek letter
rho р)
–
Positive correlation
Intelligence
Productivity
Height
Income
Academic Achievement
Value of Farm
Shoe size
Value of Home
Negative correlation
Academic Achievement
Total corn production
Practice time
Age of automobile
TV time
$ per bushel
Errors
trade-in value
Variables in Correlational Research
n
Predictor Variables:
–
n
ones in which participants' scores enable
prediction of their scores on some criterion
variable. May be thought of as independent
variables
Criterion Variables:
–
the object of the research. the variable about
which researchers seek to discover more
information. May be thought of as dependent
variables.
Critical Issues in Correlational Research
n
Development of Hypothesis
–
–
n
Selection of homogeneous groups
–
–
n
should be grounded in a theoretical framework and previous
research
caution needed for "shot-gun" research
possess variables under study
requires precise definition
Collection and analysis of data
–
–
reliability and validity of measures critical
numerous statistical procedures available - caution needed
Calculation of Correlation
a correlation is a covariance ratio
r=
∑(Zx)(Zy)
N
(Zx1 * Zy1) + (Zx2 * Zy2) + (Zx3 * Zy3) + .....
r=
x
1 +1.5
2 - .75
3 + .2
4 - 1.0
5 +1.4
6 -.10
y
xy
+1.2
- .9
+ .7
- .75
+1.2
- .30
+ 1.8
+ .68
+ .14
+ .75
+ 1.68
+ .03
5.08
N
r = 5.08 / 6 = .85
Correlation
positive: +.99
positive: +.45
negative: -.95
none: 0
Interpreting Correlation
n
Look at correlation (r) separate from
probability (p)
correlation tells you amount of relationship
positive and negative define direction of relationship
r defines amount of relationship
probability (p) tells you the odds that you observed the
relationship by chance
n
Interpretation
Coefficient
Interpretation
n
.00
.20
.40
.60
.80
to .20
to .40
to .60
to .80
to 1.00
Negligible
Low
Moderate
Substantial
High to Very High
Common Correlations
• Pearson Product Moment Correlation
– both variables are continuous
• Spearman Rank-order Correlation
– both variables are measured as rank data
• Biserial Correlation
– one variable is continuous and one is an ‘artificial’
dichotomy with an underlying normal distribution
• Point-Biserial Correlation
– one variable is continuous and one is a ‘true’
dichotomy
• Phi Coefficient
– both variables are ‘true’ dichotomies
More Correlations
• Tetrachoric Correlation
– both variables are ‘artificial’ dichotomies with underlying
normal distributions
• Polychoric Correlation
– both variables are ordinally measured with both having underlying
normal distributions
• Polyserial Correlation (rps, Dps)
– one variable is continuous and one is ordinal with an
underlying normal distribution
• Kendall Tau-b
– measures agreement between two rankings
• Kendall’s Coefficient of Concordance
– measures of the extent to which members of a set of m
distinct rank orderings of N things tend to be similar