Estimation of authenticity of statistical research.

transcript

Estimation of authenticity of statistical research

Average quadratic deviation

σ = 1

simple arithmetical method

d = V - M

genuine declination of variants from the true middle arithmetic

σ = i

method of moments

The Normal Distribution

Mean = median = mode

Skew is zero 68% of values fall

between 1 SD 95% of values fall

between 2 SDs

is needed for:1. Estimations of typicalness of the middle arithmetic (М is typical for this row, if σ is less than 1/3 of average) value.2. Getting the error of average value.3. Determination of average norm of the phenomenon, which is studied (М±1σ), sub norm (М±2σ) and edge deviations (М±3σ).4. For construction of sigmal net at the estimation of physical development of an individual.

This dispersion a variant around of average characterizes an average

quadratic deviation ( )

Coefficient of variation is the relative measure of variety; it is a percent correlation of standard deviation and arithmetic average.

Terms Used To Describe The Quality Of Measurements

Reliability is variability between subjects divided by inter-subject variability plus measurement error.

Validity refers to the extent to which a test or surrogate is measuring what we think it is measuring.

Measures Of Diagnostic Test Accuracy

Sensitivity is defined as the ability of the test to identify correctly those who have the disease.

Specificity is defined as the ability of the test to identify correctly those who do not have the disease.

Predictive values are important for assessing how useful a test will be in the clinical setting at the individual patient level. The positive predictive value is the probability of disease in a patient with a positive test. Conversely, the negative predictive value is the probability that the patient does not have disease if he has a negative test result.

Likelihood ratio indicates how much a given diagnostic test result will raise or lower the odds of having a disease relative to the prior probability of disease.

Measures Of Diagnostic Test Accuracy

Expressions Used When Making Inferences About Data

Confidence Intervals- The results of any study sample are an estimate of the true value

in the entire population. The true value may actually be greater or less than what is observed.

Type I error (alpha) is the probability of incorrectly concluding there is a statistically significant difference in the population when none exists.

Type II error (beta) is the probability of incorrectly concluding that there is no statistically significant difference in a population when one exists.

Power is a measure of the ability of a study to detect a true difference.

Multivariable Regression Methods

Multiple linear regression is used when the outcome data is a continuous variable such as weight. For example, one could estimate the effect of a diet on weight after adjusting for the effect of confounders such as smoking status.

Logistic regression is used when the outcome data is binary such as cure or no cure. Logistic regression can be used to estimate the effect of an exposure on a binary outcome after adjusting for confounders.

Survival Analysis

Kaplan-Meier analysis measures the ratio of surviving subjects (or those without an event) divided by the total number of subjects at risk for the event. Every time a subject has an event, the ratio is recalculated. These ratios are then used to generate a curve to graphically depict the probability of survival.

Cox proportional hazards analysis is similar to the logistic regression method described above with the added advantage that it accounts for time to a binary event in the outcome variable. Thus, one can account for variation in follow-up time among subjects.

Kaplan-Meier Survival Curves

Why Use Statistics?

Cardiovascular Mortality in Males

'35-'44 '45-'54 '55-'64 '65-'74 '75-'84

SMR Bangor

Roseto

Descriptive Statistics

Identifies patterns in the data Identifies outliers Guides choice of statistical test

Percentage of Specimens Testing Positive for RSV (respiratory syncytial virus)

Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun

South 2 2 5 7 20 30 15 20 15 8 4 3

North-east

2 3 5 3 12 28 22 28 22 20 10 9

West 2 2 3 3 5 8 25 27 25 22 15 12

Mid-west

2 2 3 2 4 12 12 12 10 19 15 8

Descriptive Statistics

Percentage of Specimens Testing Postive for RSV 1998-99

101520253035

Northeast

Midwest

Distribution of Course Grades

Number of Students

A A- B+ B B- C+ C C- D+ D D- F

Describing the Data with Numbers

Measures of Dispersion• RANGE • STANDARD DEVIATION• SKEWNESS

Measures of Dispersion

• RANGE • highest to lowest values

• STANDARD DEVIATION• how closely do values cluster around the

mean value• SKEWNESS

• refers to symmetry of curve

The Normal Distribution

Mean = median = mode

Skew is zero 68% of values fall

between 1 SD 95% of values fall

between 2 SDs

We take a simple random sample with replacement of 25 cards from the box as follows. Mix the box of cards; choose one at random; record it; replace it; and then repeat the procedure until we have recorded the numbers on 25 cards. Although survey samples are not generally drawn with replacement, our simulation simplifies the analysis because the box remains unchanged between draws; so, after examining each card, the chance of drawing a card numbered 1 on the following draw is the same as it was for the previous draw, in this case a 60% chance.

SIMULATION

Let’s say that after drawing the 25 cards this way, we obtain the following results, recorded in 5 rows of 5 numbers:

SIMULATION

Based on this sample of 25 draws, we want to guess the percentage of 1’s in the box. There are 14 cards numbered 1 in the sample. This gives us a sample percentage of p=14/25=.56=56%. If this is all of the information we have about the population box, and we want to estimate the percentage of 1’s in the box, our best guess would be 56%. Notice that this sample value p = 56% is 4 percentage points below the true population value π = 60%. We say that the random sampling error (or simply random error) is -4%.

SIMULATION

An experiment is a procedure which results in a measurement or observation. The Harris poll is an experiment which resulted in the measurement (statistic) of 57%. An experiment whose outcome depends upon chance is called a random experiment.

ERROR ANALYSIS

On repetition of such an experiment one will typically obtain a different measurement or observation. So, if the Harris poll were to be repeated, the new statistic would very likely differ slightly from 57%. Each repetition is called an execution or trial of the experiment.

ERROR ANALYSIS

Suppose we made three more series of draws, and the results were + 16%, + 0%, and + 12%. The random sampling errors of the four simulations would then average out to:

ERROR ANALYSIS

Note that the cancellation of the positive and negative random errors results in a small average. Actually with more trials, the average of the random sampling errors tends to zero.

ERROR ANALYSIS

So in order to measure a “typical size” of a random sampling error, we have to ignore the signs. We could just take the mean of the absolute values (MA) of the random sampling errors. For the four random sampling errors above, the MA turns out to be

ERROR ANALYSIS

The MA is difficult to deal with theoretically because the absolute value function is not differentiable at 0. So in statistics, and error analysis in general, the root mean square (RMS) of the random sampling errors is generally used. For the four random sampling errors above, the RMS is

ERROR ANALYSIS

The RMS is a more conservative measure of the typical size of the

random sampling errors in the sense that MA ≤ RMS.

ERROR ANALYSIS

For a given experiment the RMS of all possible random sampling errors is called the standard error (SE). For example, whenever we use a random sample of size n and its percentages p to estimate the population percentage π, we have

ERROR ANALYSIS

Dynamic analysis Health of people and activity of

medical establishments change in time.

Studying of dynamics of the phenomena is very important for the analysis of a state of health and activity of system of public health services.

Example of a dynamic line

Year Bed occupancy (days)

1994 340.1

1995 340.9

1996 338.0

1997 343.0

1998 341.2

1999 339.1

2000 344.2

Parameters applied for analysis of changes of a phenomenon

Rate of growth –relation of all numbers of dynamic lines to the previous level accepted for 100 %.

Parameters applied for analysis of changes of a

phenomenon

Pure gain – difference between next and previous numbers of dynamic lines.

phenomenon

Rate of gain – relation of the pure gain to previous number.

phenomenon

Parameter of visualization — relation of all numbers of dynamic lines to the first level, which one starts to 100%.

Measures of Association

Measures of Association Absolute risk

- The relative risk and odds ratio provide a measure of risk compared with a standard.

Attributable risk or Risk difference is a measure of absolute risk. It represents the excess risk of disease in those exposed taking into account the background rate of disease. The attributable risk is defined as the difference between the incidence rates in the exposed and non-exposed groups.

Population Attributable Risk is used to describe the excess rate of disease in the total study population of exposed and non-exposed individuals that is attributable to the exposure.

Number needed to treat (NNT)- The number of patients who would need to be treated to

prevent one adverse outcome is often used to present the results of randomized trials.

Relative Values

As a result of statistical research during processing of the statistical data of disease, mortality rate, lethality, etc. absolute numbers are received, which specify the number of the phenomena. Though absolute numbers have a certain cognitive values, but their use is limited.

Relative Values

In order to acquire a level of the phenomenon, for comparison of a parameter in dynamics or with a parameter of other territory it is necessary to calculate relative values (parameters, factors) which represent result of a ratio of statistical numbers between itself. The basic arithmetic action at subtraction of relative values is division.

In medical statistics themselves the following kinds of relative parameters are used:

Extensive; Intensive; Relative intensity; Visualization; Correlation.

The extensive parameter, or a parameter of distribution, characterizes a parts of the phenomena (structure), that is it shows, what part from the general number of all diseases (died) is made with this or that disease which enters into total.

Using this parameter, it is possible to determine the structure of patients according to age, social status, etc. It is accepted to express this parameter in percentage, but it can be calculated and in parts per thousand case when the part of the given disease is small and at the calculation in percentage it is expressed as decimal fraction, instead of an integer.

The general formula of its calculation is the following:

part × 100

The intensive parameter characterizes frequency or distribution.

It shows how frequently the given phenomenon occurs in the given environment.

For example, how frequently there is this or that disease among the population or how frequently people are dying from this or that disease.

To calculate the intensive parameter, it is necessary to know the population or the contingent.

General formula of the calculation is the following:

phenomenon×100 (1000; 10 000; 100 000)

environment

General mortality rate

number of died during the year × 1000

number of the population

Let’s say that after drawing the 25 cards this way, we obtain the following results, recorded in 5 rows of 5 numbers:

SIMULATION

Estimation of authenticity of statistical research.

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