Clinical Biostatistics 2

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Clinical Biostatistics 2
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Measures of the Middle

• Mean the arithmetic average of the observations
• Steps
• Add the observations to obtain their sum
• Divide the sum by the number of observations
• Weighted average
• Multiply each data value by the number of
  observations that have that value
• Add the products
• Divide the sum by the number of observations

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Measures of the Middle

• Mean
• Median
• Mode

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Measures of the Middle

• Median the middle observation or the point at
  which half the observations are smaller and the
  other half larger
• Steps
• Arrange the observations from smallest to largest
• Count in to find the middle value
• Middle value for an odd number of observations
• Mean of the two middle values for even number of
  observations
• Mode value that occurs most frequently

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What does normal distribution mean?

• Bell shaped curve
• Symmetric about the mean of the distribution
• The area under the curve is equal to 1
• Since the area under the curve is equal to 1, we
  can use the area under the curve to measure
  probabilities

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The standard normal distribution
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Distribution of observations

• How to know the shape of distribution without
  actually seeing it
• If the mean and the median are equal the
  distribution is generally symmetric
• If the mean is larger than the median, the
  distribution is skewed to the right
• If the mean is smaller than the median, the
  distribution is skewed to the left

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Skewness
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Guidelines on which measure to use

• Deciding which measure of central tendency is
  best with a given set of data
• The mean is used for numerical data and for
  symmetric (not skewed) distributions
• The median is used for ordinal data or for
  numerical data if the distribution is skewed
• The mode is used primarily for bimodal
  distribution
• The geometric mean is used primarily for
  observations measured on a logarithmic scale

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Measures of Spread

• Range
• Standard Deviation
• Coefficient of Variation
• Percentiles
• Interquartile Range

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Measures of spread

• Range difference between the smallest and
  largest observation
• Many authors use minimum and maximum values
• Standard deviation measure of the spread of
  data about their mean
• Most commonly used measure of dispersion with
  medical and health data
• Requires numerical data

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Standard deviation - importance

• Essential part of many statistical tests
• Very useful in describing the spread of
  observations about the mean
• Rules of thumb when using the standard deviation
• Regardless of how the observations are
  distributed, at least 75 of the values always
  lie between these two numbers the mean minus 2
  standard deviations and the mean plus 2 standard
  deviations
• Example mean 48.8 SD 23.8
• 48.8 2(23.8) and 48.8 2(23.8)
• 75 are between 1.2 and 96.4

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Standard deviation

• If the observation is bell shaped
• 67 of the observations are within Mean 1 SD
• 95 of the observations are within Mean 2 SD
• 99.7 of the observations are within Mean 3 SD
• Formula

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Coefficient of Variation

• Useful measure of relative spread
• Definition standard deviation divided by the
  mean times 100
• Formula

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Using different measures of dispersion

• Standard deviation is used when the mean is used
  that is with symmetric (not skewed) numerical
  data
• Percentiles and the interquartile range are used
  when
• Median is used (with ordinal data or with skewed
  numerical data)
• The mean is used, but the objective is to compare
  individual observations with a set of norms
• The interquartile range is used to describe the
  central 50 of a distribution, regardless of its
  shape
• The range is used with numerical data when the
  purpose is to emphasize extreme values
• The coefficient of variation is used when the
  intent is to compare numerical distributions
  measured on different scales

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Displaying numerical data in tables and graphs

• Stem and leaf plots
• Frequency tables
• Histograms, box plots, and frequency polygons
• Histograms
• Box plots
• Frequency polygons
• Graphs comparing two or more groups

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Summarizing nominal and ordinal data with numbers

• Nominal data
• Proportions and percentages
• Rations and rates
• Vital statistics rates
• Mortality rates
• Morbidity rates
• Adjusting rates
• Direct method
• Indirect method

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Describing relationships between two
characteristics

• Two numerical characteristics
• Correlation coefficient
• Interpretation
• Little or No relationship (0 to 0.25) or (-0 to
  -0.25)
• Fair relationship (0.25 to 0.50) or (-0.25 to
  -0.50)
• Moderate to Good relationship (0.50 to 0.75) or
  (-0.50 to --0.75)
• Very good to Excellent relationship ( more than
  0.75) or ( more than - 0.75)

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Describing relationships between two
characteristics

• Two ordinal characteristics
• Spearman rank correlation
• Interpretation
• Little or No relationship (0 to 0.25) or (-0 to
  -0.25)
• Fair relationship (0.25 to 0.50) or (-0.25 to
  -0.50)
• Moderate to Good relationship (0.50 to 0.75) or
  (-0.50 to --0.75)
• Very good to Excellent relationship ( more than
  0.75) or ( more than - 0.75)

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Describing relationships between two
characteristics

• Two nominal characteristics
• Experimental and control event rates
• The relative risk
• Absolute risk reduction and number needed to
  treat
• Odds ratio

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Population and samples

• Population large set or collection of items
  that have something in common
• Sample a subset of the population selected so
  as to be represented of the larger population
• Reasons for sampling
• Can be studied more quickly
• Study of samples is less expensive
• Study of population is impossible in most
  situations
• Sample results are more often accurate than
  population studies

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Population and samples

• If samples are properly selected, probability
  methods can be used to estimate the error in the
  resulting statistics
• Samples can be selected to reduce heterogeneity
• Methods of sampling
• Simple random sampling
• Systematic sampling
• Stratified sampling
• Cluster sampling
• Non-probability sampling

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Probability and samples

• Random assignment
• Random samples
• The Poisson distribution
• The Normal (Gaussian) distribution
• Description
• Continuous
• Smooth, bell shaped, symmetric about the mean
  (mu)
• Half the area on the left is equal to the right
• The standard normal (z) distribution