Understanding Statistics

Statistics

• A science focused on collecting, organizing, summarizing, and analyzing data to draw conclusions or answer questions.
• Considered a distinct mathematical science.
• Provides a confidence measure for conclusions.

Process of Statistics

• First, identify the research objective by detailing the question and group of interest.
• Next is data collection, gathering and measuring variables of interest systematically to answer research questions, test hypotheses, and evaluate outcomes.
• Organizing and summarizing information happens through descriptive statistics.
• Descriptive statistics uses numerical measurements, charts, graphs, and tables to give an overview of collected data.
• In the final stage, conclusions are drawn from the information, the sample data is generalized to the population using inferential statistics.
• Inferential statistics uses methods that extend sample results to the population and assess the results' reliability.

Population and Sample

• A population is the total collection of all elements of interest.
• A sample is a subgroup of the population studied in detail.

Data Collection Importance

• Empowers informed decisions.
• Helps identify problems.
• Allows development of accurate theories.
• Can back up arguments.
• Helps acquire funds.
• Increases return on assets.
• Improves quality of life.

Improper Data Collection Consequences

• Leads to inaccurate answers to research questions.
• Prevents study repetition and validation.
• Produces distorted findings that waste resources.
• Misleads researchers into fruitless investigations.
• Compromises public policy decisions.
• May cause harm to human and animal subjects.

Selection Bias

• It can occur by deliberately selecting a "representative" sample.
• Misspecifying the target population can cause selection bias.
• Selection bias can be caused failing to include the entire target population in the sampling frame (undercoverage).
• Including units not in the target population in the sampling frame (overcoverage) also has an effect.
• Other causes are multiplicity of listings in the sampling frame.
• Substituting a convenient population member for one not readily available.
• Failing to get responses from all selected samples (nonresponse).
• Allowing the sample to consist entirely of volunteers.

Qualitative Variables

• Yield categorical responses.
• Utilize words or codes to represent classes or categories.

Quantitative Variables

• Take on numerical values.
• Represent amounts or quantities.

Discrete Variables

• Quantitative variables with a finite or countable number of possible values.
• Values result from counting (e.g., 0, 1, 2, 3).

Continuous Variables

• Quantitative variables with infinite, non-countable possible values.

Levels of Measurement

• Refers to the relationship among values assigned to attributes for a variable.
• Include Nominal, Ordinal, Interval and Ratio levels.

Nominal Level

• Used to identify, name, classify, or categorize objects or events.
• Examples include method of payment, eye color, and type of school.

Ordinal Level

• Similar to nominal scales by identifying, naming, classifying or categorizing.
• They have a logical or natural order to the categories or values.
• Examples include rank of military officer and socio-economic class.

Interval Level

• Levels identify, have ordered values, and have equal distances between scales.
• Examples include temperature on Fahrenheit/Celsius thermometer and trait anxiety.

Ratio Level

• In this level data can be identified, ordered, represent equal distances between scores values, and have an absolute zero point.
• Examples include height, weight and number of words correctly spelled.

Observational Studies

• The investigator observes subjects in their natural setting.
• The researcher observes the behavior of individuals without influencing the outcome.
• The investigator identifies subjects as they occur in nature and observes some response of interest for each subject.

Experimental Studies

• Involve researcher intervention, subjects are followed over time, and the response of interest is measured.

Explanatory Variable

• A variable is manipulated or observed to determine if it causes changes in other variables.

Response Variable

• A variable that changes as a result of the explanatory variable.

Prospective Observational Studies

• The subject’s explanatory level is identified first.
• The outcome or response of interest is then observed.

Retrospective Observational Studies

• Subject’s outcomes are observed first.
• Information on any explanatory variables is then obtained.

Cross-Sectional Studies

• Observational studies gathering data at a specific moment or within a short time frame.
• Involving random subject selection from a population.
• It also involves assessment of their explanatory and response variables.
• Typically conducted retrospectively using large medical databases where patient histories are available.
• Subjects are categorized, and relationships between variables are examined.
• Generally quick and cost-effective but lack highly conclusive results.
• The effects of explanatory variables may emerge later since data is collected over a short period.
• Data collection over a short period limits the ability to provide a complete picture.

Case-Control Studies

• Typically retrospective.
• Subjects are selected on their response variable.
• Their explanatory variable (exposure factor) is then measured.
• Individuals with a specific characteristic are compared to those without it.

Cohort Studies

• Usually prospective, subjects are selected based on an explanatory variable.
• Their response outcome is observed over time.
• Cohort studies track participants for an extended period.
• Frequently used when unethical to assign certain conditions like smoking.
• They require large sample sizes when the outcome is rare to achieve significant results.
• Participant dropout is a key challenge because of the long duration.
• Cohort studies can establish temporal relationships between variables.

Experimental Unit/Subject

• A person, object, or any well-defined entity on which a treatment is applied.

Factor

• A controlled independent variable that is manipulated by the experimenter.
• It represents a general type or category of treatments.

Response Variable

• The primary outcome or measure of interest in a study or experiment.

Control Group/Non-Exposed Group

• Serves as a baseline treatment.
• Functions as comparison to other treatments.

Case/Exposed Group

• A group of experimental units where the factor is applied.

Blinding

• Used in experiments to prevent bias by ensuring that participants, researchers, or both do not know certain details about the treatment or intervention being applied.

Single-Blind Study

• Participants do not know their assigned treatment or group.
• The researchers conducting the study do know.
• Prevents participants from altering responses based on expectations or psychological effects.

Double-Blind Study

• Neither the participants nor the researchers analyzing results know who received which treatment.
• A third party has this information, eliminates participant and researcher bias for reliable, unbiased conclusions.

Steps in Designing an Experiment

• Step 1: Clearly define the research question, and ensure it provides direction for the experiment and specify the response variable and the population being studied,
• Step 2: Determine the factors affecting the response variable , including control, manipulation, and what to leave uncontrolled.
• Step 3: Determine the number of experimental units by establishing a sample size for reliable results, and considering resource limitations.
• Step 4: Determine the levels of each factor: control (keep constant or vary levels).
• Step 5: Assign experimental units randomly to treatments (randomization)to minimize bias.
• Step 6: Implement the study according to the planned design (conduct the experiment).
• Step 7: Analyze data to see if results support the initial hypothesis (test the claim).

Completely Randomized Design

• Each experimental unit is randomly assigned to a treatment group.
• Ensures equal chance distribution.

Matched-Pairs Design

• Experimental units are paired based on similar characteristics.
• Each pair receives different treatments.
• Often used in before-and-after studies or when subjects have a natural counterpart.

Randomized Block Design

• Experimental units are grouped into homogeneous blocks.
• Treatments are randomly assigned within each block.
• Helps reduce variability caused by differences within the blocks.

Observational Studies

• Cost-effective, timely, applicable to broader participant ranges due to simply observing subjects without intervention.

Designed Experiments

• Can draw conclusions about causation because they manipulate variables.

Observational Studies

• Establish associations due to not controlling variables.

Confounding

• The impacts of two or more explanatory variables are not separated.
• Relationships between explanatory and response variables may be due to other, unaccounted variables.

Lurking Variable

• An explanatory variable not considered in a study.
• Affects the value of the response variable.
• Typically related to explanatory variables considered in the study.

Confounding Variable

• An explanatory variable considered in a study.
• Its effect cannot be distinguished from a second explanatory variable.

Lurking Variables

• Not considered in the study.
• Related to both explanatory and response variables.
• It creates the apparent association between the explanatory and response variable.

Confounding Variable

• The effect does not have any association with the other explanatory variable.
• Has an effect on the response variable.

Sampling Design

• The aim is to obtain individuals for a study in a way that gets accurate information about the population.
• Too large a population makes it impossible to collect information from every community member, so sampling is used.
• Representative samples have all important qualities of the source population

Sample Size Determination

• This ensures the sample is large enough for an accurate population parameter estimate, manageable, and cost-efficient.

Confidence Level

• The level of unpredictability with a specific statistic.

Margin of Error

• The degree of error in results received from random sampling surveys.

Steps in Data Gathering

• Step 1: Set the objectives for collecting data, by identifying the purpose, establishing margin of error, confidence level and consider available funds.
• Step 2: Determine the data needed based on set objectives, including type of, design needed for the study.
• Step 3: Determine the method to be used in data gathering and define the comprehensive data collection points.
• Step 4: Design data gathering forms to be used.
• Step 5: Collect data.

Probability Sample

• Use objective chance mechanisms (randomization).
• Need a complete population listing (sampling frame).
• Selection probabilities are known.
• Referred to as random samples.
• They allow drawing valid generalizations about the universe/population.

Non-Probability Sample

• Samples are haphazard, purposeful, or volunteer-based.
• Selection probabilities are unknown.
• Should not be used for statistical inference.

Simple Random Sampling

• The most basic probability sample method.
• Assigns equal selection probabilities to each possible sample.

Systematic Random Sampling

• Selecting every kth individual from the population.
• The first individual selected corresponds to a random number between 1 to k.

Stratified Random Sampling

• Obtained by separating the population into non-overlapping groups called strata, then getting a simple random sample from each stratum.
• individuals within each stratum should be homogeneous (or similar).

Cluster Sampling

• The sample is taken from naturally occurring groups in population, and the clusters are constructed such that the sampling units are heterogeneous within the cluster and homogeneous among the clusters.

Accidental Sampling

• Is a non-probability sampling method without any system of selection, only those contacted by the researcher are surveyed.

Quota Sampling

• A non-probability sampling method with a specified number of persons of certain types included in the sample.
• The researcher draws samples from each known category within a population .

Convenience Sampling

• is a non-probability sampling method picking out people in the most convenient and fastest way to get reactions immediately.

Purposive Sampling

• the non-probability sampling technique based on criteria set by the researcher.

Judgement Sampling

• Is a non-probability sampling technique selecting samples in accordance with an expert’s judgment.

Non-Probability Sampling Use cases

• When few people are willing to be interviewed.
• When the subjects are extremely difficult to find.
• When probability sampling is too expensive to implement.
• When population elements cannot be enumerated.

Non-Sampling Error

• Errors that result from the survey process.
• Any errors that cannot be attributed to the sample-to-sample variability.

Sampling Error

• Error that results from taking one sample instead of examining the whole population.
• Errors result from using sampling to estimate.

Primary Sources

• First-hand accounts of an event or period, original thinking, and reports which are authoritative.
• Can share new information and are created at the time events occurred but can be be created later as well.

Secondary Sources

• Offer analysis, interpretation, or restatement of primary sources.
• Involve generalization, synthesis, interpretation, commentary or evaluation.

Five Methods of Collecting Primary Data

• Direct personal interviews
• Indirect/Questionnaire Method
• Focus group
• Experiment
• Observation

Open-Ended Questions

• Permit free responses that should be recorded in the respondent’s own words.
• The respondent is not given any possible answers to choose from.

Closed Questions

• Offer a list of possible options or answers from which the respondents must choose
• It is useful if the range of possible responses is known.