POPULATION: Universe of all units being studied.
If we want to study lung cancer among Indians, then the study population will be all Indians.
If we want to study lung cancer among Indian women, then the population will be all
SAMPLE: Subset of the population
RANDOM SAMPLE: Each member of the population has an EQUAL CHANCE of being chosen for the sample
SAMPLING METHODS (#5 is nonrandom)
#1 Simple Random Sample
#2 Systematic Sample
Rank 100 people by age
Beginning with the 5th person, choose every tenth
i.e. Choose the 5th,15th,25th ... 85th,95th persons
#3 Stratified Sample: Composition of the sample reflects composition of the population
Stratified Sample of 1000 Indians
Bengalis 600 (60%)
Punjabis 250 (25%)
Kashmiris 100 (10%)
Others 50 (5%)
#4 Cluster Sample (used for immunization studies)
Divide population into groups
Random sample of groups is chosen
Count every unit in each and every group selected
Divide entire city into "city blocks"
Random sample of blocks is chosen
Count every person in each city block selected
#5 Nonrandom/Convenience Sample
Interview people in a shopping mall. This is nonrandom because not everyone in the population goes shopping at the mall. People who are wheelchair-bound are less likely to visit the mall. This is also the case with people who have problems walking. Also, the mall may be too far away for those who don't have cars. If the interview is done during a weekday, people who work are unlikely to be at the mall (housewives and retirees are more likely to be interviewed on weekdays.
VALIDITY: You are actually measuring what you want to measure. Example: if IQ really measures intelligence, then IQ is high in validity. If it does not actually measure intelligence, then it is low in validity.
RELIABILITY: Refers to stability of measurement. Example: A research instrument is high in reliability if it gives consistent readings when it is used on the same subject even if the subject is measured at different times. Example of a "research instrument" is a survey questionnaire.
MEASURES OF CENTRAL TENDENCY (used to summarise a dataset)
1. Arithmetic Mean - affected by extreme values
MEASURES OF DISPERSION (used to determine how spread out a dataset is)
1. Range - difference between highest and lowest values
3. Standard Deviation - the higher the Standard Deviation, the more spread out
the data. The Standard Deviation is the square root of the Variance.
THE STANDARD DEVIATION IS A VERY IMPORTANT MEASURE - Under a Standardised Normal Curve,
68.3% of the data are found +1 or -1 standard deviation from the mean
95.5% of the data are found +2 or -2 standard deviations from the mean
99.7% are found +3 or -3 standard deviations from the mean
LEVEL OF MEASUREMENT OF DATA
1. Nominal data: qualitative, categorical data. Example: ethnicity, SEX, religion.
2. Ordinal data: Rank-ordered data. Data are grouped from low to high. But we cannot say how much lower or how much higher. Example: "low anxiety", "moderate anxiety" and "high anxiety".
3. Interval data: quantitative data. There is fixed equal interval between numbers e.g.
the difference between 10 km and 15 km is the same as the distance between 30 km
and 35 km. Examples of interval data: height, weight, temperature measured using
the Celsius scale.
4. Ratio level data: Similar to Interval Data but in addition, it has an absolute zero e.g. income, temperature measured using the Kelvin scale.
NOTE: For Ratio Data, we can use ratio level, interval level, ordinal level and nominal level statistical tests.
For Interval Data, we can use interval level, ordinal level and nominal level tests.
For Ordinal Data, we can use ordinal level and nominal level tests.
But for Nominal Data, we can only use nominal level statistical tests.
INFERENTIAL STATISTICS/STATISTICAL INFERENCE
If we are doing research on a large population, we need not study each and every individual in the population. All we need to do is choose a sample (RANDOM and REPRESENTATIVE) from the population. We can use our findings from the sample to infer (draw conclusions) about the population.
Research Hypothesis/Alternative Hypothesis: the hypothesis we wish to test. Also
called H-one and written as H1
Null Hypothesis: opposite of the Research Hypothesis. Also called H-nought and
written as H0.
Research Hypothesis - there is a statistically significant association between Smoking and Cancer
Null Hypothesis - there is no statistically significant association between Smoking and Cancer. Any association seen is due to chance
Research Hypothesis - there is a statistically significant difference between the two population means
Null Hypothesis - there is no statistically significant difference between the two population means. Any difference seen is due to chance
Significance Level/Confidence Level (denoted by α)
Usually set at 0.05 or 0.01
An α of 0.05 means we wish to test the statement "the probability of what we see occuring by chance is 5% or less i.e. p Y
Example: We test to see if X < Y
Standardised Normal Distribution: See description under "Standard Deviation"
In a normal distribution, the mean, median and mode are equal.
The curve is bell-shaped and symmetrical
The standardised normal distribution curve has a mean of zero and a standard deviation of 1.
Degree of freedom: the degree of freedom depends on the sample size or number of categories. The critical value of a statistical test changes with changes in the degree of freedom.
COMMON STATISTICAL TESTS
1. Chi-square test of goodness-of-fit,
(NOMINAL DATA - like religion)
Degree of freedom is n-1 (where n = number of categories)
2. Chi-square test of independence
Degree of freedom is (r-1) X (c-1) where r = number of rows and c = number of columns in a contingency table.
To test if there is a statistically significant association between two variables (between ethnicity and religion)
3. t-test for two independent samples
(INTERVAL DATA like height and weight).
Degree of freedom = (n1-1) + (n2-1)
To test if there is a statistically significant difference between the two population means from the two samples.
4. t-test for two matched samples
Degree of freedom is n-1 where n = the number of pairs
To test if there is a statistically significant difference between the two population means from the two matched samples.
5. If there are more than two samples, we use the ANOVA test (Analysis of Variance) or F-test
NOTE: IF THERE ARE MORE THAN TWO SAMPLES, IT IS INCORRECT TO USE THE T-TEST TO MAKE PAIRWISE COMPARISONS
Example: if there are 3 samples, it is incorrect to compare mean #1 with mean
#2, mean #1 with mean #3, mean #2 with mean #3. The ANOVA test should be done on
the three means instead.
CHOOSING A TEST
1. What is the level of measurement? Nominal, ordinal or interval?
2. How many samples? One, two or more?
3. If two samples, are they independent or paired/matched?
4. Choose the test. Make sure the assumptions of the test are not violated
ASSUMPTIONS OF CHI-SQUARE TEST OF INDEPENDENCE
1. Nominal data (ordinal data also OK)
2. 25 =< n =<250 (preferably)
3. Random sample
4. Expected value of each cell is at least 5 (if not, you should combine some of the categories)
INTERPRETING RESULTS OF CHI-SQUARE TEST
H0 is "There is no association between religion and ethnicity. Any association seen is due to chance alone"
H1 is "There is a statistically significant association between religion and ethnicity "
Reject H0 if the calculated chi-square equals or exceeds the critical value
Reject H0 if p is less than or equal to 0.05 (if testing at alpha = 0.05)
ASSUMPTIONS OF T-TEST FOR TWO INDEPENDENT SAMPLES
2. Interval data
3. Normal distribution in both groups
4. Preferably n < 30 (for each sample).
INTERPRETING RESULTS OF T-TEST
H0 is "There is no difference between mean heights of Chinese and mean heights of Japanese. Any difference seen is due to chance alone"
H1 is "There is a difference between mean heights of Chinese and mean heights of Japanese "
Reject H0 if the calculated t statistic equals or exceeds the critical value
Reject H0 if p is less than or equal to 0.05 (if testing at alpha = 0.05)
IMPORTANT: What is STATISTICALLY SIGNIFICANT may not be CLINICALLY SIGNIFICANT.
CORRELATION AND REGRESSION
Correlation: a measure of how two variables go together.
Pearson's r (also called Pearson's correlation coefficient) is a measure of linear relationship between two variables.
A value of +1 means a perfect positive linear relationship.
A value of -1 means a perfect negative linear relationship.
A value of 0 means no linear relationship.
Assumptions for using Pearson's r:
Linear relationship exists
Both variables are normally distributed
Variables measured at Interval level
It is incorrect to use r for variables measured at nominal or ordinal level
Correlations can also be nonlinear. For nonlinear correlations, we do not use Pearson's r but some other correlation coefficient.
NOTE: CORRELATION DOES NOT IMPLY CAUSATION
Just because two variables are correlated does not necessarily mean that one causes the other.
Regression: used to predict how independent variables affect a dependent variable (Y)
Simple Regression: Has only 1 dependent variable Y and 1 independent variable
Multiple Regression: Has 1 dependent variable Y but two or more independent variables X1, X2 etc
Simple regression - predict INCOME (Y) from YEARS OF EDUCATION (X)
Multiple regression - predict INCOME (Y) from YEARS OF EDUCATION (X1) and YEARS OF WORKING EXPERIENCE (X2)
(Variables measured at the nominal level such as "SEX" can also be used as independent variables in regression. They are used as "dummy variables").
r-square: An indicator of the "amount of variance of the dependent variable accounted for" by the regression equation. Also called the coefficient of determination.
The higher the r-square, the better the regression line fits the data.
Regression coefficient: If we have a regression equation Y = 0.3X1 + 4X2, then the regression coefficient of X1 is 0.3 and the regression coefficient of X2 is 4.This means that when X1 increases by 1 unit, Y will increase by 0.3Also, when X2 increases by 1 unit, Y will increase by 4 units.
Diastolic blood pressure of sample of men aged 30-50 are plotted against their age
Y = 40 + 1.5X
(Y = diastolic blood pressure, X = age)
Interpretation: For these men, each year of increase in age raises the diastolic B.P. by 1.5 mm Hg
If man is 50 years old, the predicted diastolic bp is 40 + 1.5(50) = 115 mm Hg
NOTE: It is incorrect to extrapolate in regression analysis i.e. if your sample consists of men aged 30 - 50, you should not use the regression model to predict the blood pressure of men whose ages are below 30 or above 50
WHAT TO LOOK FOR IN A GOOD REGRESSION MODEL
1. Are the dependent variable and independent variables properly "operationalised" (defined and measured)?
2. Do the relationships between the dependent variable and independent variables make sense?
3. Is the relationship between the dependent variable and each independent variable linear in nature? Do scatterplots
4. Examine the r-square. The higher the r-square, the "better" the model.
5. Examine the sign of each regression coefficient. Do they make sense?
6. Check if each of the regression coefficients are statistically significant (p =< 0.05 or p =< 0.01)
7. If there are more than one regression model, compare and contrast between them
OTHER TERMS TO KNOW
Confidence Interval: The interval within which something is likely to be found
A 95% Confidence Interval for the population mean indicates that there is a 95% probability that the population mean actually lies within that particular Confidence Interval.
Skew: If a curve is slanted to the right, it is skewed to the left.
If a curve is slanted to the left, it is skewed to the right
Nonparametric Tests: Statistical tests which make no assumptions about the parent distribution. Tests involving ranks of data are nonparametric.
Parametric Tests: Statistical tests which assume that the population distribution has a particular form e.g. a normal distribution. The t-test is a parametric test as it assumes normal distribution.
Standard Error of the Mean: We take samples from a population. For each sample, we calculate its mean. We then plot the sample means and we will get a curve. The curve will have a standard deviation. This standard deviation is the standard error of the mean. It is used to calculate confidence intervals.
The smaller the standard error of the mean, the more closely the sample mean estimates the true population mean.
Note: Originally posted by willpower at