Math in Modern World Pre Finals

Data

are raw information or facts that become useful information when organized in a meaningful way. It could be of qualitative and quantitative nature.

Data Management

is concerned with "looking after" and processing data. It involves the following:

Looking after field data sheets

Checking and correcting the raw data

Preparing data for analysis

Documenting and archiving the data and meta-data

Importance of Data Management

Ensures that data for analysis are of high quality so that conclusions are correct

Good data management allows further use of the data in the future and enables efficient integration of results with other studies.

Good data management leads to improved processing efficiency, improved data quality, and improved meaningfulness of the data.

Census

this is the procedure of systematically acquiring and recording information about all members of a given population. Researchers rarely survey the entire population for two (2) reasons: the cost is too high and the population is dynamic in that the individuals making up the population may change over time.

Sample Survey

sampling is a selection of a subset within a population, to yield some knowledge about the population of concern. The three main advantages of sampling are that (i) the cost is lower, (ii) data collection is faster, and (iii) since the data set is smaller, it is possible to improve the accuracy and quality of the data.

Experiment

this is performed when there are some controlled variables (like certain treatment in medicine) and the intention is to study their effect on other observed variables (like health of patients). One of the main requirements is the possibility of replication.

Observation study

this is appropriate when there are no controlled variables and replication is impossible. This type of study typically uses a survey. An example is one that explores the correlation between smoking and lung cancer. In this case, the researchers would collect observations of both smokers and non-smokers and then look for the number of cases of lung cancer in each group.

Characteristics of a well-designed and well-conducted survey

a. A good survey must be representative of the population.

b. To use the probabilistic results, it always incorporates a chance, such as a random number generator. Often we don't have a complete listing of the population, so we have to be careful about exactly how we are applying "chance". Even when the frame is correctly specified, the subjects may choose not to respond or may not be able to respond.

c. The wording of the question must be neutral; subjects give different answers depending on the phrasing.

d. Possible sources of errors and biases should be controlled. The population of concern as a whole may not be available for a survey. Its subset of items possible to measure is called a sampling frame (from which the sample will be selected). The plan of the survey should specify a sampling method, determine the sample size and steps for implementing the sampling plan, and sampling and data collecting.

Nonprobability sampling

is any sampling method where some elements of the population have no chance of selection or where the probability of selection can't be accurately determined. The selection of elements is based on some criteria other than randomness. These conditions give rise to exclusion bias, caused by the fact that some elements of the population are excluded. Nonprobability sampling does not allow the estimation of sampling errors. Information about the relationship between sample and population is limited, making it difficult to extrapolate from the sample to the population.

Probability Sampling

it is possible to both determine which sampling units belong to which sample and the probability that each sample will be selected.

Simple Random Sampling (SRS)

all samples of a given size have an equal probability of being selected and selections are independent. The frame is not subdivided or partitioned. The sample variance is a good indicator of the population variance, which makes it relatively easy to estimate the accuracy of results.

Systematic Sampling

relies on dividing the target population into strata (subpopulations) of equal size and then selecting randomly one element from the first stratum and corresponding elements from all other strata. A simple example would be to select every 10th name from the telephone directory, with the first selectin being random. SRS may select a sample from the beginning of the list.

Stratified Sampling

when the population embraces a number of distinct categories, the frame can be organized by these categories into separate "strata". Each stratum is then sampled as an independent sub-population. Dividing the population into strata can enable researchers to draw inferences about specific subgroups that may be lost in a more generalized random sample. Since each stratum is treated as an independent population, different sampling approaches can be applied to different strata. However, implementing such an approach can increase the cost and complexity of sample selection. Example: To determine the proportions of defective products being assembled in a factory.

Cluster Sampling

sometimes it is cheaper to 'cluster' the sample in some way (e.g. by selecting respondents from certain areas only, or certain time-periods only). Cluster sampling is an example of two-stage random sampling: in the first stage a random sample of areas is chosen; in the second stage a random sample of respondents within those areas is selected. This works best when each cluster is a small copy of the population.

Matched random sampling

in this method, there are two (2) samples in which the members are clearly paired, or are matched explicitly by the researcher (for example, IQ measurements or pairs of identical twins). Alternatively, the same attribute, or variable, may be measured twice on each subject, under different circumstances (e.g. the milk yields of cows before and after being fed a particular diet).

Characteristics of a well-designed and well-conducted experiment

a. Stating the purpose of research, including estimates regarding the size of treatment effects, alternative hypotheses, and the estimated experimental variability. Experiments must compare the new treatment with at least one (1) standard treatment, to allow an unbiased estimates of the difference in treatment effects.

b. Design of experiments, using blocking (to reduce the influence of confounding variables) and randomized assignment of treatments to subjects

c. Examining the data set in secondary analyses, to suggest new hypotheses for future study

d. Documenting and presenting the results of the study

Treatment, control groups, experimental units, random assignments and replication

a. Control groups and experimental units

To be able to compare effects and make inference about associations or predictions, one typically has to subject different groups to different conditions. Usually, an experimental unit is subjected to treatment and a control group is not.

b. Random Assignments

The second fundamental design principle is randomization of allocation of (controlled variables) treatments to units. The treatment effects, if present, will be similar within each group.

c. Replication

All measurements, observations or data collected are subject to variation, as there are no completely deterministic processes. To reduce variability, in the experiment the measurements must be repeated. The experiment itself should allow for replication itself should allow for replication, to be checked by other researchers.

Confounding

a ? variable is an extraneous variable in a statistical model that correlates (positively or negatively) with both the dependent variable and the independent variable. The methodologies of scientific studies therefore need to control for these factors to avoid a false positive (Type 1) error (an erroneous conclusion that the dependent variables are in a causal relationship with the independent variable).

Placebo and blinding

an imitation pill identical to the actual treatment pill, but without the treatment ingredients. A placebo effect is a sham (or simulated) effect when medical intervention has no direct health impact but results in actual improvement of a medical condition because the patients knew they were treated. Typically, all patients are informed that some will be treated using the drug and some will receive the insert pill, however the patients are blinded as to whether they actually received the drug or the placebo. Blinding is a technique used to make the subjects "blind" to which treatment is being given.

Blocking

is the arranging of experimental units in groups (blocks) that are similar to one another. Typically, a blocking factor is a source of variability that is not of primary interest to the experimenter. An example of a blocking factor might be the sex of a patient; by blocking on sex (that is comparing men to men and women to women), this source of variability is controlled for, thus leading to greater precision.

Completely randomized designs

are for studying the effects of one primary factor without the need to take other nuisance variables into account. The experiment compares the values of a response variable (like health improvement) based on the different levels of that primary factor (e.g., different amounts of medication). For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units (for example, using a random number generator).

Randomized block design

is a collection of completely randomized experiments, each run within one of the blocks of the total experiment. A matched pairs of design is its special case when the blocks consist of just two (2) elements (measurements on the same patient before and after the treatment or measurements on two (2) different but in some way similar patients).

Chi-Square

is used to determine whether there is significant difference between the expected value frequencies and the observed frequencies in one or more categories.

A chi-square goodness of fit test

determines if a sample data matches a population.

A chi-square test for independence

compares two (2) variables in a contingency table to see if they are related. It tests to see whether the distributions of categorical variables differ from each other.

A very small chi-square test statistic means that your observed data fits your expected data well. In other words, there is a relationship.

A very large chi-square test statistic means that the data does not fit very well. In other words, there is no relationship.