Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein,...
Transcript of Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein,...
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Variables and Descriptive Statistics
Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California, San Diego (858) 534-1818 [email protected]
UCLA, Lecture #2 of 4
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Last Class…
Scientific method – Hypothesis testing
Sampling Strategies
Types of study designs
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Purpose . . .
Variables – Variable types
– Types of data
– Scales of measurement
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Purpose…
Descriptive statistics
Categorical variables – rates, %
Continuous variables – Measures of Central Tendency – Measures of dispersion
Distributions, normal, skewed
Data display for descriptive statistics
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Introduction
Overview of statistical techniques
Includes most major types of statistical analyses needed to analyze your data
Focus – Practical considerations – Applied data analysis
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Variables
Variable = Any characteristic that can vary
Examples: Height, weight, age, behaviors, attitudes, presence of specific disease, clinical measurements, physical measures
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Variables
Independent Variable (IV) =
– Variable that is changing or manipulated – Presumed cause
Dependent Variable (DV) = – Response – Outcome
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Variables
In experiments,
– Independent Variable = Variable being manipulated by experimenter
– Dependent Variable = Is observed or measured for variation as a presumed result of the variation in the IV
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Variables
In observational studies,
– IV = variable that “logically” has some effect on the DV
– Example = Research on smoking & lung cancer
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Which is IV? DV?
The hypothesis for a study:
– There will be a significant difference in anatomy grades of DS who participated in an intensive study summer program as an undergrad compared to DS who did not participate in an intensive study program
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Which is IV? DV?
Study Description:
– Dean of faculty at a dental school is concerned about the turnover in faculty—many of them leave their positions before completing 2 years of their 5 year contract. The Dean wants to identify factors that predict commitment to teaching and decides to assess attitudes toward students and amount of money owed on loans
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Variables
Any variable can serve as the IV in one
study, and the DV or outcome in another Examples: Does use of fluoride prevent tooth decay? IV=fluoride DV=caries Does parents education level predict use of fluoride in
children? IV=education DV=fluoride
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Variables
Confounding
–A distortion in an observed relationship between an exposure & outcome brought about by a third variable
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Variables
Confounders – Associated w/ both independent & dependent
variables (eg., age in study of diabetes & AD)
– Variables that can affect or bias observed results (“Lurking variables”)
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Variables
Synergism
– The interaction of 2 causal variables so that the combined effect is greater than the sum of their effects
– Example =effect of both smoking and drinking on cognitive function is greater in combination
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Variables
Effect modification (interaction)
– The direction or strength of an association between 2 variables differs according to a third variable
– Example =coffee & cognitive function—sex modifies the association (women positive association, men no association)
– Hypertension & sex—modified by age (under 45y, men more likely to have HTN; after 45y, women more likely to have HTN)
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Types of Data
Discrete data – Categorical data – Has limited set of values – May be qualitative – Examples: eye color, blood type, gender, presence/absence of diseases, yes/no
data
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Types of Data
Continuous data – Has values that range along a continuum – Quantitative – Examples: age, body mass index, blood
pressure, # teeth – Can always take continuous data &
convert to categories
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Scales of Measurement
Nominal scales – Named categories – No particular order (1 isn’t any more
than another) – Examples: eye color, hair color, gender
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Scales of Measurement
Ordinal scales – Ordered categories – Distance between categories is unequal – Examples: 1st place, 2nd place, 3rd place; rate heath compared to others – better, the same, worse; mild, mod, severe perio disease
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Scales of Measurement
Interval (continuous) scales – Equal distance between data points – No true zero – Examples: Fahrenheit temperature – (distance 10° & 20°=distance 20° &
30°)
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Scales of Measurement
Ratio scales – Equal intervals between data points – Has true zero – Best type of scale – Examples: blood pressure, # teeth
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Scales of Measurement
Order of scales – Nominal – Ordinal – Interval – Ratio
Each successive scale has all characteristics of the previous one
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Data Analysis
Statistics = describes & presents collected data in a meaningful way
2 types of statistics – Descriptive statistics = describes the
sample, summarizes who is in sample
– Inferential Statistics = infer things about population based on sample
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Descriptive statistics
For Categorical variables – Nominal scale
Rate (% of total sample with that characteristic)
Example: total sample=150 – 15 have diabetes=10%
80 dental students 64 exercise 3x/week = 80%
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Descriptive statistics
For continuous variables Measures of central tendency
– Mean – Median – Mode
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Descriptive statistics
Measures of central tendency – Mean = average = Σ x N Where x=scores; N=total sample size Scores: 55 95 95 78 Mean= 645 = 80.625 96 8 81 63 82
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Descriptive statistics
Mean - properties – Very sensitive to small variations in
scores
– Outliers (extreme values) can cause large changes in the mean; won’t give accurate picture of the population (eg., exam scores)
– More powerful statistics use means
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Descriptive statistics
Measures of central tendency – Median = middle score, 50th percentile -Put into numerical order, middle score; if 2
middle scores, median= average of the two Scores: 55 → 55 95 63 Mean= 645 = 80.625
95 78 8 78 81 96 82 Median=81.5 81 95 63 95 82 96
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Descriptive statistics
Median – Advantages
Not as sensitive to outliers Use for describing a variable where there
are many outliers (eg., income)
– Disadvantages Statistics not as powerful
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Descriptive statistics
Measures of central tendency – Mode = Most frequently occurring score Scores: 55 → 55 95 63 Mean= 645 = 80.625
95 78 8 78 81 Median=81.5 96 82 81 95 Mode=95 63 95 82 96
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Descriptive statistics
Mode- properties – Distributions can have ≥1 mode – Bimodal distribution- distribution with 2 different peaks
2 distinct values that measurements center around example: heights of men & women
– Distributions can have no mode—all measures=frequency
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Descriptive statistics
Measures of dispersion – Another way to describe the sample – Shows how far scores are scattered
around the mean Distributions Range Variance Standard deviation
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Distributions
Normal distribution – Bell shaped
– Most data points fall in middle, w/ few very small & few very large values
– Mean, Median & Mode all occur at the same score
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Distributions
Normal distribution – Mean, Median &
Mode all occur at the same score
– Symmetrical – each half=mirror image exactly half the scores occur above and half below mean
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Distributions
Skewed to Right – looks like bell
curve w/ longer tail on right and mound pushed to left
– Most data points fall to left of middle & more very small than very large values
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Distributions
Skewed to Right
– Mean > median
– Positively skewed
– large extremes pull mean → the tail
(extremes high values)
– Median remains closer to center of the distribution
– Ex: income, CRP
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Distributions
Skewed to Left – looks like a bell
curve w/ a longer tail on left & mound pushed to right
– Most data points fall to right of middle, & there are more very large than very small values
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Distributions
Skewed to Left – Mean < median
– Negatively skewed
– large extremes pull mean → the tail
(extremes are low values)
– Median remains closer to center of the distribution
– Ex: Hormone assays
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Distributions
What if you have a skewed distribution? – Most statistics assume normality
Fairly robust to violation of assumptions But may not get accurate results if very
skewed – Data transformations-logs
Pulls in extremes Problem-logged values not clinically useful Do statistics on logged values & p based on
logs, but report unlogged means Compare results of stats w/unlogged values
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Descriptive statistics
Measures of dispersion – Describes the sample
– Shows scatter of scores around mean
Distributions Range Variance Standard deviation
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Range
Range – lowest to highest score/value – Use for continuous variables – Normally distributed, presenting mean – Example: age ranged from 18-60 months years in practice ranged from 1-25
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Range
Interquartile range (IQR) – Use w/ continuous data – Skewed data & presenting median – Divide sample into quartiles – IQR = 75th – 25th quartile – Tells where most values are located
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Descriptive statistics
Measures of dispersion – Describes the sample
– Shows scatter of scores around mean
Distributions Range Variance Standard deviation
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Variance
Shows dispersion (spread) of data points around mean
The further away the data points are from the mean, the greater the variance
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Variance
Might think the variance = average difference of each score from the mean, summed together & ÷ by total # data points or Σ (x –mean)
N but,
If normal distribution, then # data pts above mean = # data pts below mean
averaging the difference of each score from the mean=0
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Variance
Average squared deviation from the mean Computational formula:
Variance = Σ (x – mean)2
N-1 Where Σ = sum of; x = each score N=sample size or # values *Note, formula above is for sample variance; to get
population variance, use N
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Variance
Example: Community research project of teenaged mothers & their knowledge of early childhood caries
12 teen mothers in study group
Give survey to assess their knowledge & score it
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Variance
Mother Score(%) (x-48)2
1 45 9 2 45 Mean= 580=48.3% 9 3 45 12 9 variance= 4 30 Median=45% 324 2518 = 228.9 5 35 mode=45% 169 12-1 6 25 529 7 40 Range=25 – 70 64 8 50 4 9 60 variance=Σ(x-mean)2 144 10 65 N-1 289 11 70 484 12 70 484 Σ= 580 Σ= 2518
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Descriptive statistics
Measures of dispersion – Describes the sample
– Shows scatter of scores around mean
Distributions Range Variance Standard deviation
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Standard Deviation
Average deviation from the mean, ignoring the sign of the difference
The further away data points are from the mean, the greater the SD
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Standard Deviation
Computed as sq root of variance = SD=sqrt Σ (x – mean)2
N-1 For population, use N; for sample, use
N-1
w/ large sample, difference bet N or N-1 is negligible
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Standard Deviation
Mother Score(%) (x-48)2
1 45 9 2 45 Mean= 580=48.3% 9 3 45 12 9 variance= 4 30 Median=45% 324 2518 = 228.9 5 35 mode=45% 169 12-1 6 25 529 7 40 Range=25 – 70 64 SD=sqrt 228.9 8 50 4 = 15.1 9 60 variance=Σ(x-mean)2 144 10 65 N-1 289 11 70 484 12 70 SD=sqrt variance 484 Σ = 580 Σ = 2518
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Standard Deviation
SD useful to compare sets of data w/ the same mean but a different range
Example: two data sets Set A=15, 15, 15, 14, 16 Set B=2, 7, 14, 22, 30 Mean A = 15 Mean B=15 SD=sqrt 2/4=0.7 SD=sqrt 508/4=11.3 Set B-more spread out Low SD= values are not spread High SD= values very spread out
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Standard Deviation
Normal Distribution – 68% within ±1 SD
of the mean – 95% within ±2 SD
of the mean – 99% within ±3 SD
of the mean
Skewed Distribution – Eliminate scores >3 SD above or
below mean
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Data Display for Categorical data
Pie chart-- shows rates for
different categories of a nominal variable
3%
6%
22%
10%
59%
Practice type (N=175)
Government
Communitycenter/FQHC
Medical school,teaching
Hospital
Private practice
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Data Display for Descriptive Statistics
Bar graphs – Used to display
nominal or ordinal data that are discrete in nature
– Display can be horizontal
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Data Display for Descriptive Statistics
Bar graphs – Can show
comparisons of means of different groups
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<HS HS Grad Some College College Grad
Mea
n sc
ore
Education
Knowledge by Education
Comparisons of oral health knowledge scores between groups Based on educational level
![Page 59: Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California,](https://reader030.fdocuments.us/reader030/viewer/2022040800/5e3500f601fb30030c733d66/html5/thumbnails/59.jpg)
Data Display for Descriptive Statistics
Bar graphs – Data display
can be vertical
Bilat
Hyst
intact
CRP IL-6 Cortisol
![Page 60: Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California,](https://reader030.fdocuments.us/reader030/viewer/2022040800/5e3500f601fb30030c733d66/html5/thumbnails/60.jpg)
Data Display for Descriptive Statistics
Histogram – Used to display
interval or ratio scaled variables that are continuous
– Bars have = width and touch each
other indicating data are on a continuum
Age (months)
![Page 61: Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California,](https://reader030.fdocuments.us/reader030/viewer/2022040800/5e3500f601fb30030c733d66/html5/thumbnails/61.jpg)
Data Display for Descriptive Statistics
Frequency polygon – Used to display interval or
ratio scaled variables that are continuous in nature
– Shorthand way to present a histogram; use instead of histogram
– Dots are put in the center of the top of each bar and connected
![Page 62: Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California,](https://reader030.fdocuments.us/reader030/viewer/2022040800/5e3500f601fb30030c733d66/html5/thumbnails/62.jpg)
Data Display for Descriptive Statistics
Histograms (bar graphs) show comparisons between groups (cases w/Br CA vs. controls) on means of multiple continuous variables (BMD at various sites)
0
0.2
0.4
0.6
0.8
1
1.2
BMD (gm/cm2)
Cases
Controls
Site
Spine Hip F neck Ulna Radius T body
![Page 63: Variables and Descriptive Statistics...Variables and Descriptive Statistics Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California,](https://reader030.fdocuments.us/reader030/viewer/2022040800/5e3500f601fb30030c733d66/html5/thumbnails/63.jpg)
Questions????
Thank You!