8.13
Determinazione della frequenza prevista
Supponiamo di voler testare l'indipendenza tra le due variabili di una tabella di contingenza. I valori nella tabella costituiscono le frequenze osser…
Si consideri una tabella di contingenza con due variabili: consumo di alcol e mortalità per incidenti stradali, con le corrispondenti frequenze osservate.
Supponiamo che un ricercatore voglia condurre un test di indipendenza, che richiede sia le frequenze osservate che quelle attese del set di dati.
Ma in che modo il ricercatore determinerà la frequenza prevista? In primo luogo, il ricercatore presume che le due variabili siano indipendenti. Quindi, la probabilità di qualsiasi evento che coinvolga entrambe le variabili può essere calcolata utilizzando la regola di moltiplicazione per eventi indipendenti.
Il prodotto del valore di probabilità e del totale complessivo della tabella fornisce la frequenza prevista per la prima cella. Da ciò si ottiene una formula generale per la frequenza prevista, che può essere ulteriormente semplificata.
Questa formula semplificata viene quindi utilizzata per calcolare le frequenze attese delle altre celle.
Una volta calcolate tutte le frequenze previste, il ricercatore può procedere a determinare la statistica del test del chi-quadrato e condurre il test di ipotesi.
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Q1: Why do researchers assume independence between variables when calculating expected frequencies?
Assuming independence allows researchers to use the multiplication rule for independent events to calculate probabilities. When two variables are independent, they do not influence each other, so the probability of any event involving both variables equals the product of their individual probabilities. This assumption is fundamental to determining expected frequencies in a contingency table for conducting a test of independence.
Q2: What is the general formula for calculating expected frequency in a contingency table?
The expected frequency for each cell is calculated by multiplying the probability of the event by the grand total of the contingency table. This can be simplified to: expected frequency equals the row total multiplied by the column total, divided by the grand total. This formula is applied consistently across all cells to obtain complete expected frequencies for the dataset.
Q3: What minimum requirement must expected frequencies meet in a contingency table analysis?
Each expected frequency in a contingency table must be at least 5. This requirement ensures the validity and reliability of statistical tests performed on the data. If expected frequencies fall below this threshold, the results of hypothesis tests may be unreliable or invalid.
Q4: How do observed and expected frequencies differ in a test of independence?
Observed frequencies are the actual values recorded in the contingency table from collected data. Expected frequencies are theoretical values calculated assuming the two variables are independent. Comparing these two sets of frequencies allows researchers to determine whether the variables are truly independent or if a significant relationship exists between them.
Q5: What statistical test follows the calculation of expected frequencies?
After calculating expected frequencies, researchers determine the chi-square test statistic by comparing observed and expected frequencies. This statistic is then used to conduct a hypothesis test and calculate the P-value, which determines whether there is sufficient evidence to reject the independence assumption and conclude that the variables are related.
Q6: How does the multiplication rule apply to calculating expected frequencies?
The multiplication rule for independent events states that the probability of both events occurring equals the product of their individual probabilities. For expected frequencies, this rule means multiplying the individual probability values from the contingency table, then multiplying by the grand total to obtain the expected frequency for each cell.
Q7: Why are expected frequencies necessary before conducting a hypothesis test?
Expected frequencies provide the theoretical baseline needed to calculate the chi-square test statistic and assess whether observed data significantly deviates from independence. Without expected frequencies, researchers cannot quantify the difference between actual and theoretical distributions or determine the P-value required for hypothesis testing.