An introduction to an expansive and complex field Biostatistics Basics An introduction to an expansive and complex field

Common statistical terms Data Measurements or observations of a variable Variable A characteristic that is observed or manipulated Can take on different values Evidence-based Chiropractic

Statistical terms (cont.) Independent variables Precede dependent variables in time Are often manipulated by the researcher The treatment or intervention that is used in a study Dependent variables What is measured as an outcome in a study Values depend on the independent variable Evidence-based Chiropractic

Statistical terms (cont.) Parameters Summary data from a population Statistics Summary data from a sample Evidence-based Chiropractic

Evidence-based Chiropractic Populations A population is the group from which a sample is drawn e.g., headache patients in a chiropractic office; automobile crash victims in an emergency room In research, it is not practical to include all members of a population Thus, a sample (a subset of a population) is taken Evidence-based Chiropractic

Evidence-based Chiropractic Random samples Subjects are selected from a population so that each individual has an equal chance of being selected Random samples are representative of the source population Non-random samples are not representative May be biased regarding age, severity of the condition, socioeconomic status etc. Evidence-based Chiropractic

Evidence-based Chiropractic Random samples (cont.) Random samples are rarely utilized in health care research Instead, patients are randomly assigned to treatment and control groups Each person has an equal chance of being assigned to either of the groups Random assignment is also known as randomization Evidence-based Chiropractic

Descriptive statistics (DSs) A way to summarize data from a sample or a population DSs illustrate the shape, central tendency, and variability of a set of data The shape of data has to do with the frequencies of the values of observations Evidence-based Chiropractic

Evidence-based Chiropractic DSs (cont.) Central tendency describes the location of the middle of the data Variability is the extent values are spread above and below the middle values a.k.a., Dispersion DSs can be distinguished from inferential statistics DSs are not capable of testing hypotheses Evidence-based Chiropractic

Hypothetical study data (partial from book) Case # Visits 1 7 2 2 3 2 4 3 5 4 6 3 7 5 8 3 9 4 10 6 11 2 12 3 13 7 14 4 Distribution provides a summary of: Frequencies of each of the values 2 – 3 3 – 4 4 – 3 5 – 1 6 – 1 7 – 2 Ranges of values Lowest = 2 Highest = 7 etc. Evidence-based Chiropractic

Frequency distribution table Frequency Percent Cumulative % 2 3 21.4 21.4 3 4 28.6 50.0 4 3 21.4 71.4 5 1 7.1 78.5 6 1 7.1 85.6 7 2 14.3 100.0 Evidence-based Chiropractic

Frequency distributions are often depicted by a histogram Evidence-based Chiropractic

Evidence-based Chiropractic Histograms (cont.) A histogram is a type of bar chart, but there are no spaces between the bars Histograms are used to visually depict frequency distributions of continuous data Bar charts are used to depict categorical information e.g., Male–Female, Mild–Moderate–Severe, etc. Evidence-based Chiropractic

Measures of central tendency Mean (a.k.a., average) The most commonly used DS To calculate the mean Add all values of a series of numbers and then divided by the total number of elements Evidence-based Chiropractic

Formula to calculate the mean Mean of a sample Mean of a population (X bar) refers to the mean of a sample and refers to the mean of a population ΣX is a command that adds all of the X values n is the total number of values in the series of a sample and N is the same for a population Evidence-based Chiropractic

Measures of central tendency (cont.) Mode The most frequently occurring value in a series The modal value is the highest bar in a histogram Mode Evidence-based Chiropractic

Measures of central tendency (cont.) Median The value that divides a series of values in half when they are all listed in order When there are an odd number of values The median is the middle value When there are an even number of values Count from each end of the series toward the middle and then average the 2 middle values Evidence-based Chiropractic

Measures of central tendency (cont.) Each of the three methods of measuring central tendency has certain advantages and disadvantages Which method should be used? It depends on the type of data that is being analyzed e.g., categorical, continuous, and the level of measurement that is involved Evidence-based Chiropractic

Evidence-based Chiropractic Levels of measurement There are 4 levels of measurement Nominal, ordinal, interval, and ratio Nominal Data are coded by a number, name, or letter that is assigned to a category or group Examples Gender (e.g., male, female) Treatment preference (e.g., manipulation, mobilization, massage) Evidence-based Chiropractic

Levels of measurement (cont.) Ordinal Is similar to nominal because the measurements involve categories However, the categories are ordered by rank Examples Pain level (e.g., mild, moderate, severe) Military rank (e.g., lieutenant, captain, major, colonel, general) Evidence-based Chiropractic

Levels of measurement (cont.) Ordinal values only describe order, not quantity Thus, severe pain is not the same as 2 times mild pain The only mathematical operations allowed for nominal and ordinal data are counting of categories e.g., 25 males and 30 females Evidence-based Chiropractic

Levels of measurement (cont.) Interval Measurements are ordered (like ordinal data) Have equal intervals Does not have a true zero Example The Fahrenheit scale, where 0° does not correspond to an absence of heat (no true zero) In contrast to Kelvin, which does have a true zero Evidence-based Chiropractic

Levels of measurement (cont.) Ratio Measurements have equal intervals There is a true zero Ratio is the most advanced level of measurement, which can handle most types of mathematical operations Evidence-based Chiropractic

Levels of measurement (cont.) Ratio examples Range of motion No movement corresponds to zero degrees The interval between 10 and 20 degrees is the same as between 40 and 50 degrees Lifting capacity A person who is unable to lift scores zero A person who lifts 30 kg can lift twice as much as one who lifts 15 kg Evidence-based Chiropractic

Levels of measurement (cont.) NOIR is a mnemonic to help remember the names and order of the levels of measurement Nominal Ordinal Interval Ratio Evidence-based Chiropractic

Levels of measurement (cont.) Measurement scale Permissible mathematic operations Best measure of central tendency Nominal Counting Mode Ordinal Greater or less than operations Median Interval Addition and subtraction Symmetrical – Mean Skewed – Median Ratio Addition, subtraction, multiplication and division Evidence-based Chiropractic

Evidence-based Chiropractic The shape of data Histograms of frequency distributions have shape Distributions are often symmetrical with most scores falling in the middle and fewer toward the extremes Most biological data are symmetrically distributed and form a normal curve (a.k.a, bell-shaped curve) Evidence-based Chiropractic

The shape of data (cont.) Line depicting the shape of the data Evidence-based Chiropractic

The normal distribution The area under a normal curve has a normal distribution (a.k.a., Gaussian distribution) Properties of a normal distribution It is symmetric about its mean The highest point is at its mean The height of the curve decreases as one moves away from the mean in either direction, approaching, but never reaching zero Evidence-based Chiropractic

The normal distribution (cont.) Mean The highest point of the overlying normal curve is at the mean As one moves away from the mean in either direction the height of the curve decreases, approaching, but never reaching zero A normal distribution is symmetric about its mean Evidence-based Chiropractic

The normal distribution (cont.) Mean = Median = Mode Evidence-based Chiropractic

Evidence-based Chiropractic Skewed distributions The data are not distributed symmetrically in skewed distributions Consequently, the mean, median, and mode are not equal and are in different positions Scores are clustered at one end of the distribution A small number of extreme values are located in the limits of the opposite end Evidence-based Chiropractic

Skewed distributions (cont.) Skew is always toward the direction of the longer tail Positive if skewed to the right Negative if to the left The mean is shifted the most Evidence-based Chiropractic

Skewed distributions (cont.) Because the mean is shifted so much, it is not the best estimate of the average score for skewed distributions The median is a better estimate of the center of skewed distributions It will be the central point of any distribution 50% of the values are above and 50% below the median Evidence-based Chiropractic

More properties of normal curves About 68.3% of the area under a normal curve is within one standard deviation (SD) of the mean About 95.5% is within two SDs About 99.7% is within three SDs Evidence-based Chiropractic

More properties of normal curves (cont.) Evidence-based Chiropractic

Standard deviation (SD) SD is a measure of the variability of a set of data The mean represents the average of a group of scores, with some of the scores being above the mean and some below This range of scores is referred to as variability or spread Variance (S2) is another measure of spread Evidence-based Chiropractic

Evidence-based Chiropractic SD (cont.) In effect, SD is the average amount of spread in a distribution of scores The next slide is a group of 10 patients whose mean age is 40 years Some are older than 40 and some younger Evidence-based Chiropractic

Evidence-based Chiropractic SD (cont.) Ages are spread out along an X axis The amount ages are spread out is known as dispersion or spread Evidence-based Chiropractic

Distances ages deviate above and below the mean Etc. Adding deviations always equals zero Evidence-based Chiropractic

Evidence-based Chiropractic Calculating S2 To find the average, one would normally total the scores above and below the mean, add them together, and then divide by the number of values However, the total always equals zero Values must first be squared, which cancels the negative signs Evidence-based Chiropractic

Evidence-based Chiropractic Calculating S2 cont. S2 is not in the same units (age), but SD is Symbol for SD of a sample  for a population Evidence-based Chiropractic

Calculating SD with Excel Enter values in a column Evidence-based Chiropractic

SD with Excel (cont.) Click Data Analysis on the Tools menu Evidence-based Chiropractic

SD with Excel (cont.) Select Descriptive Statistics and click OK Evidence-based Chiropractic

Evidence-based Chiropractic SD with Excel (cont.) Click Input Range icon Evidence-based Chiropractic

SD with Excel (cont.) Highlight all the values in the column Evidence-based Chiropractic

SD with Excel (cont.) Click OK Check if labels are in the first row Check Summary Statistics Evidence-based Chiropractic

SD with Excel (cont.) SD is calculated precisely Plus several other DSs Evidence-based Chiropractic

Wide spread results in higher SDs narrow spread in lower SDs Evidence-based Chiropractic

Spread is important when comparing 2 or more group means It is more difficult to see a clear distinction between groups in the upper example because the spread is wider, even though the means are the same Evidence-based Chiropractic

Evidence-based Chiropractic z-scores The number of SDs that a specific score is above or below the mean in a distribution Raw scores can be converted to z-scores by subtracting the mean from the raw score then dividing the difference by the SD Evidence-based Chiropractic

Evidence-based Chiropractic z-scores (cont.) Standardization The process of converting raw to z-scores The resulting distribution of z-scores will always have a mean of zero, a SD of one, and an area under the curve equal to one The proportion of scores that are higher or lower than a specific z-score can be determined by referring to a z-table Evidence-based Chiropractic

Evidence-based Chiropractic z-scores (cont.) Refer to a z-table to find proportion under the curve Evidence-based Chiropractic

Evidence-based Chiropractic Partial z-table (to z = 1.5) showing proportions of the area under a normal curve for different values of z. Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 z-scores (cont.) Corresponds to the area under the curve in black 0.9332 Evidence-based Chiropractic