**Introduction**

Analysis of variance (ANOVA), a key statistical test in study domains such as biology, economics, and psychology, is extremely useful for analyzing datasets. It enables data comparisons between three or more groups. The fundamental distinctions between these two tests are summarized here, along with the assumptions and hypotheses that must be made regarding each type of test.

One-way ANOVA and two-way ANOVA are the two most widely utilized kinds of ANOVA. This post will go through this crucial statistical test and the distinction between the two variants of ANOVA.

**One-Way Anova vs Two-Way Anova**

There are two kinds of ANOVA: one-way (or unidirectional) and two-way (bidirectional). The number of independent variables in the analysis of variance test determines whether it is one-way or two-way. A one-way ANOVA assesses the influence of a single factor on a single response variable. It analyses if the observed differences in means across independent (unrelated) groups can be explained only by chance, or whether there are in fact any statistically significant variations between groups.

A two-way ANOVA is a variation of a one-way ANOVA. A one-way ANOVA has one independent variable that influences one dependent variable. There are two independents in a two-way ANOVA. A two-way ANOVA, for example, enables a corporation to evaluate worker productivity by using two independent variables, such as department and gender. It is used for observing the interaction of the two variables. It compares the effects of two variables at the identical time.

**Difference Between One-Way Anova and Two-Way Anova in Tabular Form**

BASIS OF COMPARISON | ONW-WAY ANOVA | TWO-WAY ANOVA |

MEANING | The one-way ANOVA hypothesis test allows you to compare the averages of three or more groups of data. | Two-way ANOVA is a hypothesis test that compares the means of three or more groups of data when two independent variables are evaluated. |

NUMBER OF INDEPENDENT VARIABLES | A single factor or independent variable is used in a one-way ANOVA. | Two independent variables and one dependent variable are used in a two-way ANOVA. |

COMPARES | One factor with three or more levels. | Multiple levels of two things have an effect. |

NUMBER OF OBSERVATION | Each group does not have to be the same. | Each group must be equal. |

DESIGN OF EXPERIMENTS | Only two principles must be met. | All three principles must be followed. |

NUMBER OF GROUP OF SAMPLES | There is just one independent variable with several groups. | Multiple groups of two factors are compared. |

EFFECT | It only has access to one variable at a time. | It has simultaneous access to two variables. As a result, it demonstrates whether there is any interaction impact among the independent variables. |

**What Is One-Way Anova?**

A one-way analysis of variance (ANOVA) is a hypothesis test that takes into account only one categorical variable or single component. It is a technique that allows us to compare the averages of three or more samples using the F-distribution. It is used to determine the difference between its various categories, each of which has multiple possible values.

The null hypothesis (H0) states that all population means are equal, but the alternative hypothesis (H1) states that at least one mean differs.

- The following assumptions underpin one method of ANOVA:
- The population where the samples are collected has a normal distribution.
- The dependent variable is measured at the interval or ratio level.
- In an independent variable, there are two or more categorical independent groups.
- Sample independence
- The population's variance is homogeneous.

**When to Use One-Way Anova?**

When you have data on one categorical variable that is independent and one quantitative dependent variable, use a one-way ANOVA. The independent variable must have a minimum of three levels (that is, at least three distinct groups or categories).

ANOVA determines if the dependent variable varies in relation to the level of the independent variable. As an example

- Social media use is your independent variable, and you allocate groups to low, medium, and high levels of social network users to see if there is a distinction in hours of sleep per night.
- Your independent variable is the soda brand, and you gather data on Coke, Pepsi, Sprite, and Fanta to see if there is a pricing difference per 100ml.
- The type of fertilizer is your independent variable, and you treat fields of crops with mixtures 1, 2, and 3 to see if there is a difference in crop production.

ANOVA's null hypothesis (H0) states that there is no difference between group means. The alternative hypothesis (Ha) is that a minimum of one group deviates significantly from the dependent variable's overall mean.

**Hypotheses of One-Way Anova**

There are two possible hypotheses in a one-way ANOVA.

- The null hypothesis (H
_{0}) states that there are no differences between groups and that means are equal (walruses weigh the same across all months). - The alternative hypothesis (H
_{1}) is that there's a difference in means and groups (walruses weigh differently in different months).

**Assumptions and Limitations of One-Way Anova**

- Normality - Normality refers to the fact that each sample is drawn from a regularly distributed population.
- Sample independence - Sample independence means that every one of them has been selected independently of the others.
- Variance equality - Variance equality requires that the variance of data in distinct groups be the same.
- Your dependent variable - in this case, "weight" - should be continuous, that is, calculated on a scale with increments (i.e., grammes, milligrams).

**Data Requirements**

Your data must match the following criteria:

- Continuous dependent variable (i.e., interval or ratio level)
- Categorical independent variable (i.e., two or more groups)
- Cases in which both the dependent and independent variables have values
- Independent samples/groups (observational independence)
- Each sample's subjects have no relationship with one another. This indicates that participants from the first group cannot also be from the second.
- No subject in the two groups can affect subjects in the other group, and neither group can influence the other.
- A population-based random collection of data
- The dependent variable's normal distribution (roughly) for every group (i.e., for each degree of the factor)
- Non-normal population distributions, particularly those with thick tails or that are strongly skewed, significantly diminish the test's power.
- A breakdown of normality in a moderate or large sample may provide quite accurate p values.
- Variance homogeneity (variances that are roughly comparable across groups)
- When this assumption is broken and sample sizes change between groups, the p-value from the overall F test is unreliable. These circumstances need the use of alternative statistics that are not predicated on equal variances across populations, including Browne-Forsythe or Welch statistics (accessible through Options in the One-Way ANOVA dialogue box).
- When this assumption fails, no matter whether the group sample sizes are comparable, the results for post hoc testing may be untrustworthy. When variances are unequal, post hoc tests that don't suggest equal variances (e.g., Dunnett's C) should be utilized.

- There are no outliers.

**What Is Two-Way Anova?**

As the name implies, two-way ANOVA is a hypothesis test in which data is classified based on two components. For example, the firm's sales are classified in two ways: first by sales made by different salespeople, and second by sales made in different locations. It is a statistical approach that allows the researcher to compare many levels (conditions) of two independent variables, each with multiple observations.

The influence of the two factors on the continuous dependent variable is investigated using two-way ANOVA. It also investigates the interactions between independent variables that influence the outcomes of the dependent variable, if any exist.

**ANOVA assumptions for two-way:**

- The population where the samples are collected has a normal distribution.
- Continuous measurement of the dependent variable.
- In two factors, there are two or more categorical independent groupings.
- The size of category independent groups should be the same.
- Observations are independent of one another.
- The population's variance is homogeneous.

**When to Use Two-Way Anova?**

When you have data on a quantitative dependent variable that is present at more than one level of two categorical independent variables, you can utilize a two-way ANOVA.

A quantitative variable indicates the number or amount of something. It can be divided to find the mean of the group.

A categorical variable describes several kinds of objects. A level is a separate category within a categorical variable.

You should have sufficient information in your data collection to calculate the average of the quantitative dependent variable for each combination of independent variable levels.

Your independent variables must be both categorical. Use an ANCOVA instead if one of the independent variables is categorical and the other is quantitative.

**Assumptions of Two-Way Anova**

Certain assumptions must be met to employ a two-way ANOVA. Two-way ANOVA assumes all the standard assumptions of a parametric difference test:

**1. Variance homogeneity (also known as homoscedasticity)**

The variation around the mean for each group under consideration should be comparable across all groups. If your data do not match this assumption, a non-parametric option, such as the Kruskal-Walli's test, may be appropriate.

**2. Observations are independent of one another.**

Your independent variables ought not to be interdependent (that is, one should not cause the other). This is impossible to verify with categorical variables; appropriate experimental design is the only way to ensure it.

Furthermore, your dependent variable ought to incorporate unique observations, which means that your data should not be aggregated among regions or individuals.

If your data do not conform to this assumption (for example, if you set up experiments inside blocks), you may include a blocking variable and/or perform a repeated-measures ANOVA.

**3. Dependent variable with a normally distributed distribution**

The dependent variable's values ought to follow a bell curve (be regularly distributed). You may try a data transformation if your data does not meet this assumption.

**How Does Two-Way Anova Test Works?**

The two-way ANOVA test is an expansion of the one-way ANOVA test, which is more speculative. An ANOVA test assesses whether a statistical process produces useful results. It essentially allows you to choose whether to reject or embrace a null hypothesis. Two factors are utilized to determine this in a two-way ANOVA test.

The two-way ANOVA test determines whether two significant variables influence the outcome or a dependent variable. The result can then be used to calculate variances and perform an f-test. The two-way ANOVA test is comparable to the two-sample t-test, except it has a lesser probability of producing type 1 errors, which could distort the results. The two-way ANOVA is versatile; it allows for comparing means and variances within and between subjects, groups, within groups, and even test groups.

A study of fertilizer kinds and planting density to produce the best agricultural production per acre is an example of applying the two-way ANOVA test. To conduct such an experiment, split the area into sections and then give each section a different type of fertilizer and planting density. When the crop matures, the amount produced in every plot is calculated. The two-way ANOVA test then reveals which fertilizer and crop density combination yields the highest harvest and how the two factors influence the outcome.

The two-way ANOVA test has numerous applications in fields such as business, public health, medicine, pharmacy, and social science.

**Main Difference Between One-Way Anova and Two Way Anova in Points**

The distinctions between one-way and two-way ANOVA are obvious on the following grounds:

- One-way ANOVA is a hypothesis test that allows us to use variance to examine the equality of three or more means at the same time. Two-way ANOVA is a statistical technique that studies the interplay between factors and influencing variables for optimal decision-making.
- In one-way ANOVA, there is a single factor or independent variable, but two-way ANOVA has two independent variables.
- Three or more levels (conditions) of one factor are compared using one-way ANOVA. Two-way ANOVA, on the other hand, examines the effect of various levels of two components.
- The number of observations in each group does not have to be the same in one-way ANOVA, but it must be the same in two-way ANOVA.
- One-way ANOVA requires only two principles of experimental design, namely replication and randomization. In contrast to Two-way ANOVA, which adheres to all three design of experiments principles of replication, randomization, and local control.

**Conclusion**

Two-way ANOVA is sometimes thought of as an enlarged form of one-way ANOVA. There are several advantages to using two-way ANOVA over one-way ANOVA, such as the ability to evaluate the effects of two factors at the same time.