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Correlation implies how well the occurrence of one phenomenon, corresponds to that of another. Causation tells us whether one phenomenon is the cause of another.A good example is to look at a disease and a symptom. Breathlessness occurs in all asthma patients. But that doesn't mean, that in all patients who experience breathlessness, the cause is asthma - it could be COPD.The proper way to deal with causation is to use Bayesian Statistics. Bayesian statistics is still a statistical method, but it doesn't just look at the frequency of an event. It looks at how well matched, an 'experience' is to a cause. For more on this, see:1.Frequentist inference2. Bayesian_inferenceThe improper use of statistics to infer causation can have serious consequences. One good example is the Sally Clark case. For a gentle introduction, see Beyond reasonable doubt. For details see:1.AppealStats.html2.Defense.html3.744.pdfRaymond Hill, a professor of mathematics at Salford university was one of the experts, who was involved in the campaign to free Sally Clark. You can see his articles here:1.Cot Death or Murder.pdf2.Reflections Article.pdf

Although highly similar, and incorrectly grouped together in SPSS, they are designed to do different things.Principal Component Analysis is designed to answer the question “Can I compress the information contained in all of these variables into a smaller package that’s easier to work with?”Reasons why you might use PCA:You have variables which are redundant in content. Suppose you had information on both shoe size and the number of inches long a foot is. They’re slightly different pieces of information, since shoe size is also contingent on foot width, but they’re highly redundant (correlated). If you had several other numeric variables about a person’s characteristics the two foot related ones would be reduced to a principal component related to foot size information. This reduction of data is really important if you need to perform a regression, but want to keep the model to a small size and/or avoid collinearity between variables (which inflates standard errors at best and screws up the magnitude and sign of the coefficients at worst).Exploratory Factor Analysis is very similar. But while PCA is a single method, EFA is a family of very similar methods. For example, you can fit an EFA model by maximum likelihood, or you can treat the variables you have as randomly sampled from a large pool of potential variables (ie, as in a random effects model, but here it’s called alpha factor analysis). There are other methods as well. Regardless, the goal is to relate the measures you have, the manifest variables, to to not-directly-observed ones you think they are measuring (the latent variables/factors).Reasons why you might use EFA:You collected a large number of questions about people’s habits, general mood, and ways of thinking. You think that each of these items likely measures some specific trait (as in a personality trait) about a person but are not completely sure what those traits are, or of exactly how many there may be. To find out you would use an exploratory factor analysis as a tool to begin building a theory/model. For example, 100 questions answered by six hundred people might yield five latent variables that you identify as the personality traits Openness to Experience, Conscientiousness, Extraversion, Agreeableness, and Neuroticism.In practice, PCA is often used as EFA. This is not strictly correct, but the practice works “well enough” and persists. Typically the principal components you get are the same as the factors, but this is not always the case. PCA also tends to be a bit over-optimistic from the perspective of EFA. In particular, PCA overestimates principal component loadings (how much each manifest variable correlates with the component), the amount of variance accounted for by the components, and underestimates the correlations between the components.This is because of the key mathematical difference between the two:Exploratory Factor Analysis only cares about the shared variance between variables, while Principal Components Analysis considers within-variable variance as well. This reflects the distinct goals, latent variable recovery vs data compression.This difference in goal can be visualized as follows: Manifest variables are thought of as “causing” a component, or “go into it” (just like files go into a zip folder to be compressed) but in factor analysis we conceive of factors as causing the manifest variables. The order is reversed. Think about it like this, personality traits are real factors that cause individual behaviors a pool of questions might ask about.There is another consequence as well. In PCA the score [math] y[/math] of an original manifest variable [math]j[/math] is due to the component loading [math]a_j[/math] times the component score [math]x_{iv}[/math] plus the residual error term [math]e[/math].[math]y_{ij} = a_j x_{iv} + e_{ij}[/math]We don’t know x_i, but want to, so the component score formula is:[math]x_{iv} = a_j^{-1} y_{ij} - e_{ij}[/math]In factor analysis the score [math]y[/math] of a manifest variable[math] _j[/math] is assumed to be caused by the score [math]x [/math]for latent variable [math]_v [/math]times the factor loading [math]a [/math]of manifest variable [math]_j[/math] onto [math]_v[/math]. Instead of a single residual error term[math] e[/math] as before, we have the communal error term [math]h[/math] for measure [math]_j [/math]times the unique error term for person [math]_i[/math] on measure [math]_j[/math].[math]y_{ij} = a_j x_{iv} + h_{jv} z_{ij} [/math]Since we don’t know [math]x_iv[/math] but want to, the formula becomes:[math]x_{iv} = a_j^{-1} y_{ij} - h_{jv} z_{ij} [/math]But as I said, the unique variances of manifest variables are not accounted for in EFA, so we don’t know what proper number to plug into the equation! This leads to the factor score indeterminacy problem where there end up being nearly infinite ways to create factor scores consistent with the model. In practice, there are a small number of methods that are accepted that give pretty close results. Still, if the generated factor scores are used in estimation, significance testing, or hypothesis testing the subtle differences might affect the conclusion!The final difference is that factor analysis includes more than exploratory analysis. Once you think you have a good understanding of what the latent variables (Factors) are you can build a structural equation model of how each of those manifest variables relate to the latent variables. This is known as confirmatory factor analysis.Some sources of interest:Edit: I just happened to stumble across this pdf today and it explains the answer to your question in far more depth than I did but is still fairly accessible, I highly recommend reading it: https://www.personality-project.org/r/book/Chapter6.pdfhttp://www.real-statistics.com/multivariate-statistics/factor-analysis/factor-scores/https://pareonline.net/pdf/v10n7.pdfhttps://pareonline.net/pdf/v14n20.pdfhttp://www.statpower.net/Steiger%20Biblio/SteigerSchonemann78b.pdf******Note that in my formulas I only show the coefficient for a single component/factor and a single manifest variable in the formula for simplicity. In reality, if you have three factors, the score for an observed variable would be[math]y_{ij} = a_1 x_{i1} + a_2 x_{i2} + a_3 x_{i3} + \sum_j h_{jv} z_{ij}[/math].and the score for a factor, if there are 5 manifest variables (in reality you’d have way more than this, like the 100 personality questions in the EFA example)[math]x_{iv} = a_{v1}^{-1} y_{i1} + a_{v2}^{-1} y_{i2} + a_{v3}^{-1} y_{i3} + a_{v4}^{-1} y_{i4} + a_{v5}^{-1} y_{i5} - \sum_j h_{jv} z_{ij} [/math]Or just this for short[math]X = A^{-1}Y - HZ [/math]Or even shorter,[math]X = A^{-1}Y [/math] as typically written (ie, see pg 173 of the Steiger chapter linked above expressing X for component scores).

After a job interview, if asked “Do you have any questions?” always say “ Yes, is there anything about my application that concerns you?”.Runny nose? Push your tongue against the top of your mouth and push a finger between your eyebrows for about 20 - 30 seconds.Can't think of what to get someone for their birthday or as a gift? Tell them you already got them a gift and to guess what it is. They will tell you some ideas of what they want.You can win someone's heart by praising or complimenting her/him in a group, it works like magic.Want to get rid of bad breath? Brushing your teeth is important but it's more important to brush your tongue. That's where bad breath starts.Set an alarm 30 minutes before you go to bed.Uninstall every addicting app.Ask yourself, “Do I need it?”. If no, then uninstall.If yes, even then uninstall. Almost all social media apps can be used in the browser. This action will easily reduce addiction.I’d also recommend you use Quora online rather than the app; it works a lot betterWanna drive safer?. Pretend every other driver is stupid. That’ll raise your cautionsUse Banana peel to shine your shoes.It's natural and no risk of staining your carpet.Not sure if a pair of shoes fits, & no time to try them on? Your foot is the same size as your forearm.Want a great haircut? Tell the barber that you have an important interview next week. Giving them a purpose ensures a good job.If you want to read a book but don’t wanna pay for a paperback or e-book, search for an online PDF!If you’re watching YouTube and you wanna pause the video, try the letter “K” and it’ll pause the video without going down to the comment sectionIf you are working on a laptop and can’t stop looking at your phone, hide it behind your laptop.Running low on battery? Put your phone on aeroplane mode and it’ll charge much Faster.Having a stomach ache? Lie down on your left side, and rub your stomach in clockwise circles. It should help.If you know you are going to be in a crowded space, take note of the exits.If you’re feeling stressed or have no motivation, clean your room.Source