Mathematical principles and statistics Flashcards
What is a logarithm?
A mathematical function for transforming large or complex numbers for the purpose of simplification
The log is the power to which the base must be raised to produce the number
x = logbase (number)
So for log10 (1000), log = 3
Log usually refers to log10
ln refers to the natural log, where the base is Euler’s number = 2.71828
Log rules:
* logxy = logx + logy
* logx/y = logx - logy
* log1/x = -logx
* logxʸ = y logx
* Any log of 1 = 0
* Any log its own base = 1
What is pH and the pH scale?
pH is a measurement of the acidity of a substance measured on a logarithmic scale
pH = -log [H+]
When the concentration of H+ is represented in mol/L
A change in pH by one whole unit represents a change in the [H+] by a factor of 10
@pH 7, [H+] = 10-7
@pH 6, [H+] = 10-6
@pH 8, [H+] = 10-8
What is differentiation?
A mathematical tool that allows the rate of change between one variable and other variables to be formulated - i.e. calculate the gradient of a non-linear relationship
(Note: Calculating the gradient of a straight line = change in Y / change in X)
What is integration?
The process of calculating the multiplication sum of two variables where one variable is changing. It may be described as finding the area under the curve
The process essentially involves dividing a curve into an infinite number of tiny rectangles, the sum of which is the area under the curve
What is an exponential?
A condition where the rate of change of a variable at any point is proportional to the value of the variable at that time
I.e. the larger the quantity grows, the faster it grows
Positive exponential: y = e^x
What is the half life (t1/2)?
The time taken for the value of an exponential function to decrease by half
An exponential process is said to have completed after five half lives - at this point 96.875% of the process has occurred
What is the time constant (𝜏)?
The time it would take for a negative exponential process to complete if the initial rate of change was maintained throughout
OR the time taken for the value of an exponential function to fall to 37% of its previous value
An exponential process is said to have completed after three time constants - at this point 94.9% of the process has occurred
What is a wash-in curve?
Physiological build-up negative exponential
The curve is a negative exponential because the rate of increase in y is decreasing exponentially as time increases
An example is the ‘wash-in’ of a volatile anaesthetic representing the rate of rise of alveolar concentration, shown on a time vs FA / FI graph
What is a wash-out curve?
Physiological negative exponential
An example is seen in drug elimination
What is tear-away?
Clinical tear-away positive exponential
An example is seen in the early stages of bacterial replication
Standard deviation
= Square root of the variance
In a normal distribution:
* 1 SD - 68% of data
* 2 SD - 96% of data
* 3 SD - 99% of data
Absolute risk reduction and number needed to treat
ARR = Control event rate - Treatment event rate
NNT = 1 / ARR
E.g. 8% have a stroke > treatment introduced and 4% have a stroke
ARR = 0.08 - 0.04 = 0.04
NNT = 1 / 0.04 = 25
Types of error
Type 1 error = false positive = rejecting a null hypothesis that is actually true
Type 2 error = false negative = accepting a null hypothesis that is actually false
Types of data
Quantitative
* Discrete - whole number values
* Continuous - can take on any value within a range
Qualitative
* Nominal - items belong to different categories with no relationship
* Ordinal - items with an inherent order, may be assigned numbers but no mathematical relationship
* Dichotomous - binary yes / no
When to use different statistical tests
Quantitative and parametric (normal distribution)
* 2 groups - paired or unpaired t-test
* >2 groups - paired or unpaired ANOVA
Quantitative and non-parametric
* 2 groups paired - Wilcoxon signed-rank
* 2 groups unpaired - Mann-Witney U
* >2 groups paired - Friedman
* >2 groups unpaired - Kruskal-Wallis
Qualitative
* Small sample size - Fisher’s exact
* Large sample size - Chi square
